Information Geometry of Asymmetric Interaction Matrices

Abstract

Asymmetric interaction matrices encode the linear coupling structure and directed interaction patterns that arise in mathematical models of complex networks across ecology, finance, and machine learning, yet their geometric structure as points on a statistical manifold has received comparatively little systematic treatment. This paper develops a rigorous information-geometric framework for the manifold ${\mathcal{M}}_n^{+}$ of real $n\times n$ interaction matrices whose symmetric part is negative definite—equivalently, the matrices satisfying the numerical stability condition $\omega \left(A\right)={\lambda }_{\max }\left(S\left(A\right)\right)<0$. The symmetric part $S\left(A\right)=(A+A^{\mathsf{T}})/2$ and the skew-symmetric part $K\left(A\right)=(A-A^{\mathsf{T}})/2$ correspond, respectively, to the metric structure and the torsion of the induced statistical manifold. We construct the natural augmented Riemannian metric g on ${\mathcal{M}}_n^{+}$ as the sum of the Fisher–Rao pullback metric through $-S(·)$ and a Frobenius term on $K(·)$, derive explicit formulae for the sectional curvature in the mixed symmetric–skew plane, and prove that the sectional curvature vanishes if and only if A is normal. The central theoretical result is a curvature-mediated stability theorem: a Fisher–Rao stability margin, derived from the precision representative of A, provides a sharp, computationally accessible certificate for the asymptotic stability of the linear dynamical system $\dot{\mathbf{x}}=A\mathbf{x}$, with the instability boundary lying at infinite Fisher–Rao distance. We further establish an information-geometric reformulation of May’s stability criterion for random ecological networks, a curvature-based covariance regularisation scheme for financial correlation matrices, and a Jacobian stability bound for deep neural networks. All the main results are illustrated with explicit $3\times 3$ and $4\times 4$ numerical examples.

1. Introduction

Interaction matrices arise wherever one system influences another: they encode predator–prey relations in ecological food webs, directed cash flows in financial networks, synaptic weights in neural circuits, and state-transition rates in non-equilibrium physical systems. The overwhelming majority of such matrices are asymmetric—the influence of entity j on entity i is not, in general, reciprocated—and this asymmetry is not a perturbation of some underlying symmetric structure but an intrinsic, structurally essential property. Yet, the standard toolkit of matrix analysis is dominated by results for symmetric or Hermitian matrices, and the geometry of the space of asymmetric interaction matrices has remained comparatively unexplored.

We use the term complex network in its established technical sense [1,2]: a graph whose coupling structure is heterogeneous (the edge weights vary across the network and follow a broad, often heavy-tailed, distribution rather than taking a single value), directed (the adjacency matrix is non-symmetric, $A_{ij}\ne A_{ji}$), and typically large, with non-trivial higher-order organisation such as in hubs, communities, or motifs. This is to be contrasted with a simple or regular network—for example, a lattice, a complete graph, or a regular ring—in which the coupling is homogeneous ($A_{ij}$ takes one of a few fixed values), the adjacency matrix is symmetric or circulant, and the spectrum is known in closed form. The two cases differ sharply in the present context. A simple network with symmetric coupling has a normal interaction matrix, for which the spectral and numerical abscissae coincide ($\alpha =\omega$) and the non-normality defect vanishes; its stability is fully captured by the eigenvalues alone. A complex network, by contrast, generically has a non-normal interaction matrix, for which $\alpha <\omega$ and transient amplification can be large even under spectral stability—precisely the regime through which the information-geometric quantities of this paper provide information that the eigenvalues do not. It is in this restricted, technical sense, rather than a claim about all physical systems in nature, that we speak of interaction matrices in models of complex networks.

Information geometry, founded by Rao [3] and Chentsov [4] and developed into a comprehensive theory by Amari and Nagaoka [5], provides a framework for endowing families of probability distributions with a canonical Riemannian metric—the Fisher–Rao metric—and a pair of dual affine connections. The central insight is that the Fisher information matrix, originally conceived as a measure of statistical distinguishability, is simultaneously the natural Riemannian metric on the statistical manifold and encodes in its curvature the departure of the family from exponential form. Murray and Rice [6] and Shima [7] extended this programme to general Hessian and dually flat structures; Nielsen and Bhatia [8] developed the matrix-specific theory with applications to covariance geometry.

The connection between information geometry and interaction matrices emerges from a simple observation: an asymmetric matrix $A\in {\mathbb{R}}_{\mathrm{c}}^{n\times n}$ with a negative-definite symmetric part $S\left(A\right)=(A+A^{\mathsf{T}})/2$—the matrices that are stable in the numerical-abscissa sense—uniquely determines a Gaussian distribution on ${\mathbb{R}}_{\mathrm{c}}^n$ with precision matrix $-S\left(A\right)\succ 0$. The parameter space of such matrices therefore inherits a natural statistical manifold structure, with the Fisher–Rao metric measuring how much two interaction matrices differ in their induced distributional effects. The skew-symmetric part $K\left(A\right)=(A-A^{\mathsf{T}})/2$ does not affect the induced distribution but encodes the directional structure of the interaction—the net information flow between entities—and enters the geometry as torsion.

A precise terminological distinction is in order, as the two notions are mathematically distinct, and we follow the terminology of Amari and Nagaoka [5] precisely.

Statistical manifold. A statistical manifold is a smooth manifold $\mathcal{S}$ whose points are probability distributions: one fixes a parametric family $\left\{p\right(·;\xi ):\xi \in \mathsf{\Xi }\}$ and treats the parameter space $\mathsf{\Xi }$ as a differentiable manifold. At this level, only the smooth structure is required; no metric or connection is yet assumed.

Information manifold. An information manifold is a statistical manifold additionally equipped with the following information-theoretic structures of information geometry: the Fisher information metric

$$g_{ij}\left(\xi \right)={\mathbb{E}}_{p(·;\xi )}\left[{\displaystyle \frac{\partial \log p}{\partial {\xi }^i}}{\displaystyle \frac{\partial \log p}{\partial {\xi }^j}}\right],$$

(1)

a dual pair of $\alpha$-connections $({\nabla }^{\left(\alpha \right)},{\nabla }^{(-\alpha )})$ satisfying the compatibility relation

$$Xg(Y,Z)=g\left({\nabla }_X^{\left(\alpha \right)}Y,Z\right)+g\left(Y,{\nabla }_X^{(-\alpha )}Z\right),$$

(2)

and a divergence such as the Kullback–Leibler divergence $\mathrm{KL}(p\parallel q)$, whose asymmetry $\mathrm{KL}(p\parallel q)\ne \mathrm{KL}(q\parallel p)$ is encoded by the non-self-duality of the pair. Consequently, every information manifold is a statistical manifold, but a statistical manifold becomes an information manifold only once the Fisher metric and dual connections are endowed; the containment is strict and runs in this direction. We give the full development of this structure, as well as the precise family of distributions used in this paper, in Section 2.6.

The role of these notions here. The underlying information manifold of the present paper is the space of zero-mean Gaussians $\{\mathcal{N}(0,P^{-1}):P\in {\mathrm{SPD}}_n\}$ parameterised by the precision matrix P, carrying the Fisher–Rao metric $g^F$ and the Amari $\alpha$-connections (Proposition 3); this is a genuine information manifold in the sense above. The object ${\mathcal{M}}_n^{+}$ of the interaction matrices studied in this paper is not itself a statistical manifold—its points are matrices, not distributions, and the assignment $A\mapsto \mathcal{N}(0,-S{\left(A\right)}^{-1})$ is many-to-one, collapsing each skew-symmetric fibre $\{A:S\left(A\right)=S_0\}$ to a single Gaussian. Rather, ${\mathcal{M}}_n^{+}$ is a Riemannian manifold whose metric is obtained by pullback from the Gaussian information manifold along the precision map $P:{\mathcal{M}}_n^{+}\to {\mathrm{SPD}}_n$, $A\mapsto -S\left(A\right)$, together with a flat Frobenius term on the skew-symmetric directions, which carry no distributional content. We are therefore careful throughout this paper not to call ${\mathcal{M}}_n^{+}$ a statistical or information manifold; it is the total space of a trivial bundle over the Gaussian information manifold ${\mathrm{SPD}}_n$, and it is in this precise sense that the framework “projects a system with asymmetric interactions onto an information-geometric manifold.”

The stability of the linear system $\dot{x}=Ax$ is determined by the spectral abscissa $\alpha \left(A\right)={\max }_i\mathrm{Re}\left({\lambda }_i\left(A\right)\right)$: the system is asymptotically stable if and only if $\alpha \left(A\right)<0$. Classical results from matrix analysis, in particular the numerical abscissa inequality $\alpha \left(A\right)\le \omega \left(A\right)≔{\lambda }_{\max }\left(S\left(A\right)\right)$ (see, e.g., Horn and Johnson [9], Chapter 5), connect the spectral abscissa to the symmetric part of A. For symmetric matrices $(\alpha =\omega )$, this reduces to the familiar eigenvalue criterion. For asymmetric matrices, the gap $\omega \left(A\right)-\alpha \left(A\right)\ge 0$ reflects non-normality and transient amplification, phenomena that are central to the ecologically motivated instability analysis by May [10] and the subsequent refinements by Allesina and Tang [11].

The present paper develops these connections systematically within a unified information-geometric framework. The main contributions are as follows. First, we define and study the Riemannian manifold ${\mathcal{M}}_n^{+}$ of interaction matrices with a negative-definite symmetric part (the numerically stable matrices), equipping it with a natural augmented metric g that is non-degenerate on the full parameter space. Second, we compute the sectional curvature of $({\mathcal{M}}_n^{+},g)$ in closed form and prove that it vanishes if and only if A is normal. Third, we establish a stability certificate—the geodesic distance from A to the instability boundary $\partial {\mathcal{M}}_n^{+}$—and show that it provides a computationally accessible lower bound on the stability margin $-\omega \left(A\right)$. Fourth, we apply this framework to three canonical domains, namely ecological community matrices, financial covariance matrices, and neural network Jacobians, in each case recovering known stability criteria as geometric consequences and deriving new information-geometric bounds.

The remainder of this paper is organised as follows. Section 2 collects the necessary preliminaries from matrix analysis and information geometry. Section 3 surveys existing methods for stability analysis of asymmetric interaction matrices—Lyapunov stability theory, spectral and numerical abscissa methods, random matrix theory, pseudospectral analysis, and Stein covariance estimation—and identifies the structural limitations that motivate the new geometric approach. Section 4 constructs the Riemannian manifold ${\mathcal{M}}_n^{+}$ and derives the augmented metric and curvature formulae, and Section 5 develops the curvature-mediated stability theory. The geometric framework is then completed before any application is presented: Section 6 develops the geodesic structure, exponential map, and parallel transport; Section 7 develops the dual connections and $\alpha$-geometry; Section 8 establishes the pseudospectral certificates; and Section 9 presents the computational algorithms. With the complete geometry in hand, Section 10 applies the framework to ecology, finance, neural networks, advection–diffusion operators, and Leontief input–output systems, and Section 11 provides explicit numerical validation. Section 12, Section 13 and Section 14 present extended numerical studies, the discussion, and the conclusions.

2. Mathematical Preliminaries

2.1. Decomposition of Asymmetric Matrices

Every real matrix $A\in {\mathbb{R}}_{\mathrm{c}}^{n\times n}$ admits a unique orthogonal decomposition with respect to the Frobenius inner product ${\langle X,Y\rangle }_F=\mathrm{tr}\left(X^{\mathsf{T}}Y\right)$ as follows:

$$A=S\left(A\right)+K\left(A\right),S\left(A\right)≔\frac{1}{2}(A+A^{\mathsf{T}})\in {\mathrm{Sym}}_n,K\left(A\right)≔\frac{1}{2}(A-A^{\mathsf{T}})\in {\mathrm{Skw}}_n,$$

(3)

where ${\mathrm{Sym}}_n=\{X\in {\mathbb{R}}_{\mathrm{c}}^{n\times n}:X=X^{\mathsf{T}}\}$ and ${\mathrm{Skw}}_n=\{X\in {\mathbb{R}}_{\mathrm{c}}^{n\times n}:X=-X^{\mathsf{T}}\}$ are the spaces of the symmetric and skew-symmetric matrices, respectively. These spaces are orthogonal complements in $({\mathbb{R}}_{\mathrm{c}}^{n\times n},{\langle ·,·\rangle }_F)$: ${\langle S\left(A\right),K\left(B\right)\rangle }_F=0$ for all $A,B$. The decomposition (3) satisfies ${∥A∥}_F^2={∥S\left(A\right)∥}_F^2+{∥K\left(A\right)∥}_F^2$, a consequence of the Pythagorean theorem for orthogonal complements.

A matrix A is called normal if $A{A}^{\mathsf{T}}=A^{\mathsf{T}}A$. Normality is equivalent to the commutativity of its symmetric and skew-symmetric parts: A is normal if and only if $\left[S\right(A),K(A\left)\right]≔S\left(A\right)K\left(A\right)-K\left(A\right)S\left(A\right)=0$ [9]. This equivalence is fundamental: the commutator $[S,K]$ is symmetric (indeed ${[S,K]}^T={\left(SK\right)}^T-{\left(KS\right)}^T=K^T{S}^T-S^T{K}^T=KS-SK·(-1)$; using $S^T=S$, $K^T=-K$ gives ${[S,K]}^T=-KS+SK=[S,K]$) and measures the degree of non-normality.

2.2. Asymptotic Stability

Definition 1

(Asymptotic (Hurwitz) Stability). A matrix $A\in {\mathbb{R}}_{\mathrm{c}}^{n\times n}$ is asymptotically stable (equivalently, Hurwitz stable) if every eigenvalue has a strictly negative real part: $\alpha \left(A\right)≔{\max }_i\mathrm{Re}\left({\lambda }_i\left(A\right)\right)<0$. Equivalently, the solution of the linear system $\dot{\mathbf{x}}=A\mathbf{x}$ satisfies $\mathbf{x}\left(t\right)=e^{At}{\mathbf{x}}_0\to 0$ as $t\to \infty$ for every initial condition ${\mathbf{x}}_0\in {\mathbb{R}}_{\mathrm{c}}^n$.

The Lyapunov stability theorem provides the following classical algebraic characterisation: A is Hurwitz if and only if there exists a symmetric positive-definite matrix $P\succ 0$ satisfying $AP+P{A}^{\top }+Q=0$ for some $Q\succ 0$, in which case $V\left(x\right)=x^{\top }Px$ is a strict global Lyapunov function with $\dot{V}=-x^{\top }Qx<0$ along trajectories [12].

Asymptotic stability is equivalent to an explicit exponential-decay estimate, which we record because it is the quantitative form used throughout this paper. The following statements are equivalent for $A\in {\mathbb{R}}_{\mathrm{c}}^{n\times n}$:

(i)

$\alpha \left(A\right)<0$ (spectral condition);

(ii)

$x\left(t\right)=e^{At}{x}_0\to 0$ as $t\to \infty$ for every $x_0$;

(iii)

There exist constants $M\ge 1$ and $\beta >0$ such that ${∥e^{At}∥}_2\le M{e}^{-\beta t}$ for all $t\ge 0$;

(iv)

There exists $P\succ 0$ with $AP+P{A}^{\top }+I=0$, the unique solution being the controllability Gramian $P={\int }_0^{\infty }{e}^{At}{e}^{A^{\top }t}dt$.

The decay rate in (iii) may be taken to be any $\beta <-\alpha \left(A\right)$, and when A is normal, one may take $M=1$ and $\beta =-\alpha \left(A\right)=-\omega \left(A\right)$. For a non-normal A, the constant M may exceed unity, allowing the transient growth ${\sup }_{t\ge 0}{∥e^{At}∥}_2>1$ that the non-normality defect of Section 5 quantifies; this is the precise sense in which spectral stability ($\alpha <0$) does not by itself preclude large finite-time amplification. The equivalence (i) ⇔ (iv) is the Lyapunov characterisation specialised to $Q=I$, and the solution P is exactly the matrix whose Fisher–Rao geometry drives the stability theory developed below.

This paper develops three nested sufficient certificates for asymptotic stability: the spectral criterion $\alpha \left(A\right)<0$ (Definition 2), the numerical-abscissa criterion $\omega \left(A\right)<0$ (which implies $\alpha \left(A\right)<0$ since $\alpha \le \omega$), and the information-geometric criterion, namely the stability margin $m^F\left(A\right)=\log {\lambda }_{\min }(-S\left(A\right))$ (finite and with the boundary at infinite distance, equivalent to $\omega \left(A\right)<0$; see Proposition 5).

2.3. Spectral Abscissa and Numerical Abscissa

Definition 2

(Spectral and Numerical Abscissae). For $A\in {\mathbb{R}}_{\mathrm{c}}^{n\times n}$, the spectral abscissa is $\alpha \left(A\right)≔{\max }_i\mathrm{Re}\left({\lambda }_i\left(A\right)\right)$, where ${\lambda }_i\left(A\right)$ are the eigenvalues of A. The numerical abscissa (also called the logarithmic norm) is $\omega \left(A\right)≔{\lambda }_{\max }\left(S\left(A\right)\right)$.

The fundamental inequality $\alpha \left(A\right)\le \omega \left(A\right)$ holds for every $A\in {\mathbb{R}}_{\mathrm{c}}^{n\times n}$ [9,12]. For normal matrices, the two quantities coincide, as the following classical result establishes.

Theorem 1

(Normal Equivalence of Abscissae). If $A\in {\mathbb{R}}_{\mathrm{c}}^{n\times n}$ is normal, then $\alpha \left(A\right)=\omega \left(A\right)$.

Proof. 

Since A is normal over ${\mathbb{R}}_{\mathrm{c}}$, there exists a unitary matrix $U\in {\mathbb{C}}_{\mathrm{c}}^{n\times n}$ such that $U^{*}AU=\mathsf{\Lambda }=\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_n)$. Then, $U^{*}S\left(A\right)U=S\left(\mathsf{\Lambda }\right)=\mathrm{diag}(\mathrm{Re}\left({\lambda }_1\right),\dots ,\mathrm{Re}\left({\lambda }_n\right))$, so the eigenvalues of $S\left(A\right)$ are exactly $\left\{\mathrm{Re}\left({\lambda }_i\right)\right\}$. Hence, $\omega \left(A\right)={\lambda }_{\max }\left(S\left(A\right)\right)={\max }_i\mathrm{Re}\left({\lambda }_i\right)=\alpha \left(A\right)$. □

Remark 1

(Non-Normal Gap). When A is not normal, a strict inequality where $\alpha \left(A\right)<\omega \left(A\right)$ can occur. The extreme case is illustrated by $A_{ϵ}=\left(\begin{array}{cc}-1& 1/ϵ\\ 0& -1\end{array}\right)$ for small $ϵ>0$: eigenvalues are $-1,-1$ so $\alpha \left(A_{ϵ}\right)=-1<0$ (stable), yet $\omega \left(A_{ϵ}\right)={\lambda }_{\max }\left(\begin{array}{cc}-1& 1/\left(2ϵ\right)\\ 1/\left(2ϵ\right)& -1\end{array}\right)=-1+1/\left(2ϵ\right)\to +\infty$ as $ϵ\to 0$. The system is globally stable, yet its numerical abscissa is arbitrarily large, reflecting severe transient amplification governed by non-normality.

2.4. Computability over the Computable Reals

Because the computable real numbers have Lebesgue measure zero, almost every real number is non-computable, and one might worry that statements about real-valued matrices are problematic from a computability standpoint. We resolve this by working over the computable reals ${\mathbb{R}}_{\mathrm{c}}$ throughout the framework of computable analysis [13,14,15], and we show that the entire geometric construction of this paper is closed within this setting.

Definition 3

(The Computable Reals). A real number x is computable if there exists a Turing machine that, on input $k\in \mathbb{N}$, outputs a rational $q_k$ with $|x-q_k|\le 2^{-k}$; that is, x admits a Turing-computable Cauchy sequence of rationals with a known modulus of convergence. The set of computable reals is denoted ${\mathbb{R}}_{\mathrm{c}}$.

The set ${\mathbb{R}}_{\mathrm{c}}$ is a computable ordered field: it is a subfield of $\mathbb{R}$ containing $\mathbb{Q}$, it is real-closed, and the field operations $+,-,\times ,÷$ together with $\sqrt{·}$ on non-negative elements are all computable in the sense of Definition 3 [13]. Crucially, ${\mathbb{R}}_{\mathrm{c}}$ contains every computable irrational number—$\sqrt{2}$, e, $\pi$, and all algebraic numbers—so it is far richer than $\mathbb{Q}$ and, unlike $\mathbb{Q}$, is closed under the transcendental operations (matrix exponential and logarithm) needed for the geometry below. A function $f:{\mathbb{R}}_{\mathrm{c}}^m\to {\mathbb{R}}_{\mathrm{c}}^k$ is computable if a Turing machine transforms any fast-converging rational approximation of the input into a fast-converging rational approximation of the output, uniformly.

Definition 4

(Computable SPD Matrices). The space of computable symmetric positive-definite matrices is

$$\mathrm{SPD}(n,{\mathbb{R}}_{\mathrm{c}})≔\{A\in \mathrm{Sym}(n,{\mathbb{R}}_{\mathrm{c}}):A\succ 0\},$$

(4)

the positive-definite matrices, all of whose entries lie in ${\mathbb{R}}_{\mathrm{c}}$, where $\mathrm{Sym}(n,{\mathbb{R}}_{\mathrm{c}})$ denotes symmetric matrices over ${\mathbb{R}}_{\mathrm{c}}$.

Proposition 1

(Closure of $\mathrm{SPD}(n,{\mathbb{R}}_{\mathrm{c}})$ under the Geometric Operations). The set $\mathrm{SPD}(n,{\mathbb{R}}_{\mathrm{c}})$ is closed under the operations required by the information-geometric framework of this paper. Specifically, for $A,B\in \mathrm{SPD}(n,{\mathbb{R}}_{\mathrm{c}})$, the following all lie in, or map into, $\mathrm{SPD}(n,{\mathbb{R}}_{\mathrm{c}})$ (respectively, ${\mathbb{R}}_{\mathrm{c}}$), and are computable as functions of their arguments: matrix product $AB$ and inverse $A^{-1}$; the eigenvalues ${\lambda }_1\left(A\right),\dots ,{\lambda }_n\left(A\right)$ and the numerical abscissa ${\lambda }_{\max }\left(S\right)$; the matrix square root $A^{1/2}$ and its inverse $A^{-1/2}$; the matrix exponential $\exp \left(A\right)$ and logarithm $\log \left(A\right)$; the determinant and log-determinant; and the Fisher–Rao distance $d^F(A,B)$ of Equation (7).

Proof. 

Matrix multiplication and inversion are rational functions of the entries (with non-vanishing determinant on ${\mathrm{SPD}}_n$), and are hence computable and ${\mathbb{R}}_{\mathrm{c}}$-valued. For the eigenvalues, the ordered list ${\lambda }_1\left(A\right)\le \cdots \le {\lambda }_n\left(A\right)$ of a symmetric matrix, counted with multiplicity, is a computable function of A: by Weyl’s inequality, each ${\lambda }_i$ is 1-Lipschitz in the operator norm of A, so it is uniformly continuous with an explicit modulus, and the characteristic polynomial has ${\mathbb{R}}_{\mathrm{c}}$ coefficients computable from the entries; the increasingly ordered real roots of a real-rooted polynomial with computable coefficients are computable even across coincidences of roots, since the ordered-root map is continuous and effectively locally uniformly approximable [13,15]. (It is the spectral projections, not the eigenvalues, that can fail to be computable where multiplicities change; the quantities used here—${\lambda }_{\max }$, ${\lambda }_{\min }$, det, $\log \det$, and the functional calculus below—depend only on the eigenvalue list and on A itself, not on a choice of eigenbasis.) In particular, each ${\lambda }_i\left(A\right)\in {\mathbb{R}}_{\mathrm{c}}$ and ${\lambda }_{\max }\left(S\right),{\lambda }_{\min }\left(S\right)\in {\mathbb{R}}_{\mathrm{c}}$. The matrix square root, inverse square root, exponential, and logarithm of an element of $\mathrm{SPD}(n,{\mathbb{R}}_{\mathrm{c}})$ are computable because each is the limit, with explicit error bounds, of a uniformly convergent sequence of polynomials (respectively, Padé approximants) in A on the norm-bounded spectrum—e.g., $\log A={\sum }_{k\ge 1}{\displaystyle \frac{{(-1)}^{k+1}}{k}}{(A-I)}^k$ after scaling A into the convergence region, and exp, $\sqrt{·}$ likewise—so they are computable as matrix functions of A directly, without reference to an eigenbasis, and return symmetric matrices with ${\mathbb{R}}_{\mathrm{c}}$ entries, positive-definite when the scalar function is positive. The log-determinant is ${\sum }_i\log {\lambda }_i\in {\mathbb{R}}_{\mathrm{c}}$, and the Fisher–Rao distance $d^F(A,B)={∥\log \left(A^{-1/2}B{A}^{-1/2}\right)∥}_F$ is a composition of these computable maps, and is hence computable and ${\mathbb{R}}_{\mathrm{c}}$-valued. □

Theorem 2

(Computable Geodesic Completeness). For $A,B\in \mathrm{SPD}(n,{\mathbb{R}}_{\mathrm{c}})$, the affine-invariant Fisher–Rao geodesic

$$\gamma \left(t\right)=A^{1/2}{\left(A^{-1/2}B{A}^{-1/2}\right)}^t{A}^{1/2},t\in {\mathbb{R}}_{\mathrm{c}},$$

(5)

is a computable function of t, and $\gamma \left(t\right)\in \mathrm{SPD}(n,{\mathbb{R}}_{\mathrm{c}})$ for every $t\in {\mathbb{R}}_{\mathrm{c}}$. Consequently, $\left(\mathrm{SPD}(n,{\mathbb{R}}_{\mathrm{c}}),g^F\right)$ is geodesically complete, and the information manifold restricted to computable entries, ${\mathcal{M}}_n^{+}\left({\mathbb{R}}_{\mathrm{c}}\right)\cong \mathrm{SPD}(n,{\mathbb{R}}_{\mathrm{c}})\times \mathrm{Skw}(n,{\mathbb{R}}_{\mathrm{c}})$, is a computable Riemannian manifold: charts, tangent vectors, the metric, geodesics, the exponential and logarithmic maps, and the curvature are all computable.

Proof. 

The matrix power $M^t=\exp \left(t\log M\right)$ for $M=A^{-1/2}B{A}^{-1/2}\in \mathrm{SPD}(n,{\mathbb{R}}_{\mathrm{c}})$ is computable in both M and $t\in {\mathbb{R}}_{\mathrm{c}}$ by Proposition 1 (computability of log, scalar multiplication by t, and exp), and $M^t\in \mathrm{SPD}(n,{\mathbb{R}}_{\mathrm{c}})$ since the exponential of a symmetric ${\mathbb{R}}_{\mathrm{c}}$-matrix is positive-definite with ${\mathbb{R}}_{\mathrm{c}}$ entries. Conjugating by $A^{1/2}\in \mathrm{SPD}(n,{\mathbb{R}}_{\mathrm{c}})$ preserves both positive-definiteness and ${\mathbb{R}}_{\mathrm{c}}$-valuedness, so $\gamma \left(t\right)\in \mathrm{SPD}(n,{\mathbb{R}}_{\mathrm{c}})$ for all $t\in {\mathbb{R}}_{\mathrm{c}}$. Since $\gamma$ is defined for all $t\in {\mathbb{R}}_{\mathrm{c}}$ and never leaves $\mathrm{SPD}(n,{\mathbb{R}}_{\mathrm{c}})$—the completeness of ${\mathbb{R}}_{\mathrm{c}}$ as an ordered field ensures the eigenvalues ${\lambda }_i{\left(M\right)}^t$ remain positive and finite for every finite t—the space is geodesically complete. The remaining geometric data are computable by Proposition 1 and the explicit formulae (18) and (31), applied entrywise over ${\mathbb{R}}_{\mathrm{c}}$. □

Theorem 2 shows that no expressive power is lost by restricting to computable entries: the full Hadamard manifold structure of Section 6, the curvature formulae of Section 4, and every certificate computed by the algorithms of Section 9 survive verbatim over ${\mathbb{R}}_{\mathrm{c}}$. We therefore adopt the convention, throughout this paper, that ${\mathbb{R}}_{\mathrm{c}}$ and ${\mathbb{C}}_{\mathrm{c}}$ denote the computable reals ${\mathbb{R}}_{\mathrm{c}}$ and computable complexes ${\mathbb{C}}_{\mathrm{c}}$; all matrices, eigenvalues, connections, geodesics, and stability certificates are computable objects in the sense above. The associated connections are computable connections: their Christoffel symbols (Equation (33)) are computable functions of the base point and tangent vectors since they are built from the computable operations of Proposition 1. Working over $\mathbb{Q}$ would not suffice—$\mathrm{SPD}(n,\mathbb{Q})$ is not geodesically complete, as $M^t$ generically leaves $\mathbb{Q}$ even for a rational M and t—whereas ${\mathbb{R}}_{\mathrm{c}}$ is both rich enough to contain all the irrational quantities that arise and closed under every operation the geometry requires. At the level of numerical implementation, the algorithms operate in IEEE-754 double-precision arithmetic, a finite subset of the dyadic rationals contained in ${\mathbb{R}}_{\mathrm{c}}$, so the computable framework is consistent with the floating-point computations reported in Section 11.

2.5. The Fisher–Rao Metric on Symmetric Positive-Definite Matrices

Let ${\mathrm{SPD}}_n≔\{X\in {\mathrm{Sym}}_n:X\succ 0\}$ denote the cone of symmetric positive-definite matrices. As a smooth manifold, ${\mathrm{SPD}}_n$ has dimension $n(n+1)/2$ and tangent space $T_X{\mathrm{SPD}}_n\cong {\mathrm{Sym}}_n$ at every point X. The family of centred Gaussian distributions $\mathcal{N}(0,X^{-1})$ parameterised by the precision matrix $X\in {\mathrm{SPD}}_n$ carries the canonical Fisher–Rao metric [3,5]

$$g_X^F(U,V)=\frac{1}{2}\mathrm{tr}\left(X^{-1}U{X}^{-1}V\right),U,V\in T_X{\mathrm{SPD}}_n\cong {\mathrm{Sym}}_n.$$

(6)

This metric is the unique (up to scaling) Riemannian metric on ${\mathrm{SPD}}_n$ that is invariant under the congruence action $X\mapsto P^{T}XP$ for all $P\in \mathrm{GL}(n,{\mathbb{R}}_{\mathrm{c}})$, a result due to Chentsov [4]. The geodesic distance in $({\mathrm{SPD}}_n,g^F)$ is given by

$$d^F(X,Y)={\left({\displaystyle \sum _{i=1}^n}{\log }^2\left({\mu }_i\right)\right)}^{1/2},$$

(7)

where ${\mu }_1,\dots ,{\mu }_n$ are the eigenvalues of $X^{-1/2}Y{X}^{-1/2}$ [16,17,18]. The sectional curvature of $({\mathrm{SPD}}_n,g^F)$ is non-positive and equals $-{\displaystyle \frac{1}{2}}{∥[U,V]∥}_F^2/({∥U∥}_F^2{∥V∥}_F^2-{\langle U,V\rangle }_F^2)$ for tangent vectors $U,V$ at $X=I$ [6,16].

2.6. Differences Between Statistical and Information Manifolds

Because the distinction between a statistical manifold and an information manifold is central to the correct interpretation of this paper, we state it precisely, following Amari and Nagaoka [5].

Definition 5

(Statistical and Information Manifolds). A statistical manifold is a smooth manifold $\mathcal{S}$ together with a diffeomorphism on a parametric family of probability distributions $\left\{p\right(·;\xi ):\xi \in \mathsf{\Xi }\}$, with $\mathsf{\Xi }\subseteq {\mathbb{R}}^d$ open, and $p(·;\xi )$ depending smoothly on ξ and all densities being mutually absolutely continuous. An information manifold is a statistical manifold $\mathcal{S}$ equipped, in addition, with

(a)

The Fisher information metric $g=\left(g_{ij}\right)$ of (1);

(b)

A dual pair of α-connections $({\nabla }^{\left(\alpha \right)},{\nabla }^{(-\alpha )})$ satisfying (2);

(c)

A contrast (divergence) function $D(·\parallel ·)$, canonically the Kullback–Leibler divergence, inducing g and $({\nabla }^{\left(\alpha \right)},{\nabla }^{(-\alpha )})$ through its derivatives.

This relationship is a strict, one-directional containment.

Proposition 2

(Hierarchy of the Two Notions). Every information manifold is a statistical manifold. The converse fails: a statistical manifold becomes an information manifold only after it is endowed with the Fisher metric and the dual α-connections of Definition 5(a)–(b). In particular, possession of the underlying smooth family of distributions does not by itself determine the information-geometric structure.

Proof. 

The forward implication is immediate from Definition 5, since an information manifold is, by definition, a statistical manifold with extra structure. For the converse, the smooth structure of $\mathcal{S}$ does not single out a metric: any positive-definite $(0,2)$-tensor field defines a Riemannian metric, and the Fisher metric (1) is one such specific choice determined by likelihood, not by the manifold topology. Likewise, the $\alpha$-connections are additional data; distinct contrast functions on the same family yield distinct dual pairs. Hence, the assignment of $(g,{\nabla }^{\left(\alpha \right)},{\nabla }^{(-\alpha )})$ is meant for genuine extra structure, and the containment is strict [5] (§3.1–3.2). □

The probability distributions chosen in this paper. The statistical manifold underlying the present work is the family of non-degenerate zero-mean multivariate Gaussian distributions on ${\mathbb{R}}^n$,

$$\mathcal{S}=\left\{p(\mathbf{x};P)={\left(2\pi \right)}^{-n/2}{\left(\det P\right)}^{1/2}\exp \left(-\frac{1}{2}{\mathbf{x}}^{\top }P\mathbf{x}\right):P\in {\mathrm{SPD}}_n\right\},$$

(8)

parameterised by the precision (inverse covariance) matrix $P={\mathsf{\Sigma }}^{-1}$. This is a smooth $\left({\displaystyle \genfrac{}{}{0pt}{}{n+1}{2}}\right)$-dimensional family, hence a statistical manifold with coordinate $\xi =P$. It is promoted to an information manifold by endowing it with the following: (a) the Fisher metric, which for this family is exactly the affine-invariant Fisher–Rao metric $g_P^F(U,V)=\frac{1}{2}\mathrm{tr}\left(P^{-1}U{P}^{-1}V\right)$ on ${\mathrm{SPD}}_n$ (derived in Section 4); (b) the Amari $\alpha$-connections, computed in Proposition 3; and (c) the Kullback–Leibler divergence between Gaussians, which in precision coordinates is the Bregman divergence of the log-partition function $\psi \left(P\right)=-\frac{1}{2}\log \det P$. These three structures are precisely the information-theoretic data of Definition 5, and their explicit forms for the Gaussian family are the subject of the remainder of this section.

Why ${\mathcal{M}}_n^{+}$ is neither, and how it is linked to $\mathcal{S}$. The interaction matrix space ${\mathcal{M}}_n^{+}$ is not a statistical manifold: its points are matrices, and the map $A\mapsto p(·;-S(A\left)\right)\in \mathcal{S}$ is many-to-one, since every A in the skew fibre $\{A^{\prime }:S\left(A^{\prime }\right)=S\left(A\right)\}$ yields the same Gaussian. The link between ${\mathcal{M}}_n^{+}$ and the information manifold $\mathcal{S}$ is the precision projection $P:{\mathcal{M}}_n^{+}\to {\mathrm{SPD}}_n$, $A\mapsto -S\left(A\right)$, which is a surjective submersion; the metric on ${\mathcal{M}}_n^{+}$ used throughout is the pullback $P^{*}{g}^F$ along this map, augmented by a flat Frobenius metric on the skew-symmetric fibre directions (Section 4). Thus, ${\mathcal{M}}_n^{+}$ inherits its symmetric-direction geometry from the Gaussian information manifold, while the skew directions—which do not change the associated distribution—carry no information-theoretic content and are given the trivial flat metric. This is the precise and consistent sense in which the asymmetric dynamical data of A is “projected onto” an information-geometric manifold.

2.7. Statistical Manifolds and Dual Connections

Recall from Definition 5 that the information-geometric data on a statistical manifold is a metric together with a dual pair of connections. A pair of torsion-free affine connections ∇ and ${\nabla }^{*}$ on a Riemannian manifold $(\mathcal{S},g)$ is dual with respect to g if, for any vector fields $X,Y,Z$,

$$X{\langle Y,Z\rangle }_g={\langle {\nabla }_{X}Y,Z\rangle }_g+{\langle Y,{\nabla }_X^{*}Z\rangle }_g.$$

(9)

The Levi-Civita connection ${\nabla }^{\left(0\right)}={\displaystyle \frac{1}{2}}(\nabla +{\nabla }^{*})$ is the self-dual element. Amari’s $\alpha$-connections ${\nabla }^{\left(\alpha \right)}$ interpolate between the mixture connection ${\nabla }^{(-1)}$ and the exponential connection ${\nabla }^{(+1)}$ [5]. For an exponential family, the e-geodesics (straight lines in the ${\nabla }^{(+1)}$-sense) correspond to mixture paths in natural parameters, while the m-geodesics correspond to affine paths in mean parameters. The curvature of ${\nabla }^{\left(\alpha \right)}$ measures the departure of the family from flatness (dually flat structure); its vanishing characterises exponential and mixture families [7,8].

We now make explicit how this structure arises for the Gaussian precision family that underlies the present paper in three steps.

Step 1 (the exponential family). The Gaussian family $\{\mathcal{N}(0,S^{-1}):S\in {\mathrm{SPD}}_n\}$ parameterised by the precision matrix S is a full exponential family. In canonical form, $p(\mathbf{x};\theta )=\exp (\langle \theta ,T(\mathbf{x})\rangle -\psi (\theta \left)\right)$ with natural parameter $\theta =S/2$, sufficient statistic $T\left(\mathbf{x}\right)=-\mathbf{x}{\mathbf{x}}^{\top }$, and log-partition function $\psi \left(\theta \right)=-\frac{1}{2}\log \left|2\theta \right|+\frac{n}{2}\log \left(2\pi \right)$. The Fisher information of this family is the Fisher–Rao metric $g^F$ of Section 4 below.

Step 2 (the canonical connections). The following standard fact fixes the connections referred to throughout this paper.

Proposition 3

(Dual Connections of the Gaussian Precision Family). For the Gaussian family $\{\mathcal{N}(0,S^{-1}):S\in {\mathrm{SPD}}_n\}$:

(i) 

The exponential connection ${\nabla }^{(+1)}$ is flat, with affine coordinates given by the natural parameters S (i.e., ${\nabla }^{(+1)}$-geodesics are straight lines $S\left(t\right)=(1-t)S_0+t{S}_1$);

(ii) 

The mixture connection ${\nabla }^{(-1)}$ is flat, with affine coordinates given by the mean parameters $\mathsf{\Sigma }=S^{-1}$ (i.e., ${\nabla }^{(-1)}$-geodesics are straight lines $\mathsf{\Sigma }\left(t\right)=(1-t){\mathsf{\Sigma }}_0+t{\mathsf{\Sigma }}_1$);

(iii) 

${\nabla }^{(+1)}$ and ${\nabla }^{(-1)}$ are dual with respect to $g^F$ in the sense of (9), and their midpoint ${\nabla }^{\left(0\right)}=\frac{1}{2}({\nabla }^{(+1)}+{\nabla }^{(-1)})$ is the Levi-Civita connection of $g^F$.

Proof. 

Parts (i)–(iii) represent the specialisation of the general theory of dually flat exponential families [5] (Ch. 2–3) to the Gaussian model, using the canonical form of Step 1. The flatness of ${\nabla }^{(+1)}$ in natural coordinates and of ${\nabla }^{(-1)}$ in mean (= expectation) coordinates is the defining property of a dually flat statistical manifold; the duality relation and the Levi-Civita midpoint follow from the $\alpha$-connection identities ${\nabla }^{\left(\alpha \right)}=\frac{1+\alpha }{2}{\nabla }^{(+1)}+\frac{1-\alpha }{2}{\nabla }^{(-1)}$. □

Step 3 (extension to ${\mathcal{M}}_n^{+}$). On the product manifold ${\mathcal{M}}_n^{+}\cong {\mathrm{SPD}}_n\times {\mathrm{Skw}}_n$ (constructed in Section 4), the $\alpha$-connection is defined component-wise: it is the Amari $\alpha$-connection of Proposition 3 on the ${\mathrm{SPD}}_n$ factor, and the flat Euclidean connection (all Christoffel symbols zero) on the ${\mathrm{Skw}}_n$ factor. The full dual structure on ${\mathcal{M}}_n^{+}$ is therefore the direct product ${\nabla }_{{\mathcal{M}}_n^{+}}^{\left(\alpha \right)}={\nabla }_{{\mathrm{SPD}}_n}^{\left(\alpha \right)}\oplus {\nabla }_{{\mathrm{Skw}}_n}^{\left(0\right)}$, developed in detail in Section 7.

3. Classical and Existing Methods for Stability Analysis

The stability of a linear dynamical system $\dot{\mathbf{x}}=A\mathbf{x}$ is a foundational problem in applied mathematics, and a rich toolkit of classical methods has been developed to certify, estimate, and monitor it. This section surveys the principal existing approaches, organised by the mathematical object they use to characterise stability. For each approach, we highlight both its strengths and the structural gap that motivates the information-geometric framework developed in subsequent sections.

3.1. Lyapunov Stability Theory

The classical approach to certifying asymptotic stability is the method of Lyapunov [12]. A matrix $A\in {\mathbb{R}}_{\mathrm{c}}^{n\times n}$ is Hurwitz stable (i.e., $\alpha \left(A\right)<0$) if and only if there exists a symmetric positive-definite matrix $P\succ 0$ satisfying the algebraic Lyapunov equation

$$PA+A^{\mathsf{T}}P+Q=0,Q\succ 0.$$

(10)

For a given Q, the unique solution $P=P(A,Q)={\int }_0^{\infty }{e}^{A^{\mathsf{T}}t}Q{e}^{At}dt$ is positive-definite if and only if A is Hurwitz. The function $V\left(\mathbf{x}\right)={\mathbf{x}}^{T}P\mathbf{x}$ is then a global Lyapunov function with $\dot{V}=-{\mathbf{x}}^{T}Q\mathbf{x}<0$ along system trajectories.

The standard choice $Q=I$ gives $PA+A^{\mathsf{T}}P+I=0$, whose solution $P={\int }_0^{\infty }{e}^{A^{\mathsf{T}}t}{e}^{At}dt$ encodes the observability Gramian of the pair $(A,I)$. The eigenvalues of P measure the energy of the system’s impulse response: $\mathrm{tr}\left(P\right)={\int }_0^{\infty }{∥e^{At}∥}_F^{2}dt$ is the total $H_2$ norm squared of A.

• Strengths. Lyapunov theory provides exact stability conditions with no approximation, is applicable to nonlinear systems via generalisation, and yields explicit energy-decay bounds. The Lyapunov equation (10) can be solved in $O\left(n^3\right)$ operations using the Bartels–Stewart algorithm [19].
• Limitations. The solution P depends on both A and Q, and the choice of Q is not canonical. Different choices of Q yield qualitatively different Lyapunov functions and different stability margins. More fundamentally, Lyapunov theory does not provide a natural geometry on the space of interaction matrices—it gives a certificate for each fixed A but offers no principled way to compare two interaction matrices, to regularise A towards stability, or to quantify how far A is from the instability boundary in a metric sense. The present paper addresses all three gaps by placing the Lyapunov equation within an information-geometric framework (Section 5, Theorem 8).

3.2. Spectral and Numerical Abscissa Methods

The spectral abscissa $\alpha \left(A\right)={\max }_i\mathrm{Re}\left({\lambda }_i\left(A\right)\right)$ is the tightest stability certificate: $\alpha \left(A\right)<0$ is necessary and sufficient for asymptotic stability. However, computing eigenvalues of a general asymmetric matrix requires $O\left(n^3\right)$ operations and is sensitive to perturbations when A is non-normal (defective or near-defective). For large systems, the full eigenvalue decomposition is both expensive and numerically fragile.

A computationally cheaper and more robust alternative is the numerical abscissa $\omega \left(A\right)={\lambda }_{\max }\left(S\left(A\right)\right)$ of Definition 2, which satisfies $\alpha \left(A\right)\le \omega \left(A\right)$ and equals $\alpha \left(A\right)$ if and only if A is normal [9]. The numerical abscissa is computable from a symmetric eigenvalue decomposition, which is backward-stable, parallelisable, and available in $O\left(n^2\right)$ memory. It therefore serves as the primary stability check in many large-scale applications in ecology [10], control [12], and numerical methods [20].

The gap $\omega \left(A\right)-\alpha \left(A\right)\ge 0$ measures non-normality: how much the asymmetric interactions ($K\left(A\right)\ne 0$) allow transient amplification before long-run decay. Non-normal systems can exhibit large initial amplification—potentially of orders of magnitude—before eventually converging, a phenomenon termed transient instability [21]. Quantifying this gap rigorously requires either pseudospectral tools (Section 3.4) or the information-geometric commutator norm introduced in Section 5.

• Limitations. The numerical abscissa is conservative as a stability margin: a matrix with $\omega \left(A\right)=-0.01$ and $\alpha \left(A\right)=-2.0$ is classified as “barely stable” by $\omega$ but is robustly stable by $\alpha$. There is no classical method to tighten this bound without computing eigenvalues. Moreover, the numerical abscissa gives no information about how close A is to the instability boundary in any geometric sense; it only indicates whether A is on the correct side.

3.3. Random Matrix Theory and May’s Stability Criterion

A landmark result in theoretical ecology is May’s stability criterion [10]: for a random community matrix A with off-diagonal entries drawn i.i.d. from a distribution with mean zero and variance ${\sigma }^2$, self-regulation $d>0$ (so that $A_{ii}=-d$), and connectivity C (fraction of non-zero off-diagonal entries), the system is almost surely stable if and only if $\sigma \sqrt{nC}<d$. This is a consequence of Girko’s circular law [22] and its refinements [23]: the empirical spectral distribution of a large i.i.d. random matrix concentrates on a disk of radius $\sigma \sqrt{n}$ in the complex plane, so that $\alpha \left(A\right)\approx -d+\sigma \sqrt{nC}$.

May’s criterion was refined by Allesina and Tang [11], who show that the structure of the community matrix (predator–prey, mutualistic, or competitive) affects the stability threshold via the correlation between $A_{ij}$ and $A_{ji}$. Specifically, the circular law must be replaced by an elliptic law when $\mathbb{E}\left[A_{ij}{A}_{ji}\right]=\rho {\sigma }^2\ne 0$, giving the threshold $\sigma \sqrt{nC(1+|\rho \left|\right)}<d$ for predator–prey structure ($\rho <0$).

• Strengths. Random matrix theory gives sharp, asymptotically exact stability thresholds for large random networks, and the thresholds depend only on a few summary statistics (${\sigma }^2$, n, C, $\rho$).
• Limitations. Random matrix results are asymptotic in n and assume a distributional structure that real networks may not satisfy. They give no information about specific matrices—only about ensemble averages. More importantly, they say nothing about the transient dynamics: two matrices with the same $\alpha \left(A\right)$ can have wildly different transient amplification depending on their non-normality. The information-geometric framework of this paper addresses this by separately tracking the spectral margin (via $\omega$) and the transient risk (via $\mathcal{C}\left(A\right)$).

3.4. Pseudospectral Analysis

Pseudospectral methods [21] quantify the sensitivity of the spectrum to perturbation. The $\epsilon$-pseudospectrum of A is

$${\mathsf{\Lambda }}_{\epsilon }\left(A\right)=\{z\in \mathbb{C}:{\sigma }_{\min }(zI-A)\le \epsilon \}=\bigcup _{{\parallel E\parallel }_2\le \epsilon }\mathrm{spec}(A+E),$$

(11)

and the pseudospectral abscissa ${\alpha }_{\epsilon }\left(A\right)=\sup \{\mathrm{Re}\left(z\right):z\in {\mathsf{\Lambda }}_{\epsilon }\left(A\right)\}$ measures how far the $\epsilon$-pseudospectrum extends into the right half-plane. A stable matrix A (with $\alpha \left(A\right)<0$) is robustly stable if ${\alpha }_{\epsilon }\left(A\right)<0$ for some $\epsilon >0$.

The Kreiss constant [21] provides a related but computationally more tractable certificate as follows:

$$\mathcal{K}\left(A\right)=\underset{x>0}{\sup }x{∥{(xI+A)}^{-1}∥}_2.$$

(12)

The Kreiss matrix theorem guarantees ${\sup }_{t\ge 0}{∥e^{tA}∥}_2\le e\mathcal{K}\left(A\right)$, so that a small Kreiss constant implies moderate transient growth. However, computing $\mathcal{K}\left(A\right)$ accurately requires evaluating ${∥{(xI+A)}^{-1}∥}_2$ over a grid of x-values, each requiring $O\left(n^3\right)$ operations.

• Strengths. Pseudospectral methods are the gold standard for quantifying non-normal transient growth and are applicable to both continuous- and discrete-time systems. They reveal stability structure that eigenvalues alone cannot capture.
• Limitations. Computing pseudospectra accurately for a large n is expensive: standard algorithms require $O\left(n^3\right)$ per grid point in the complex plane, and the full pseudospectral portrait may require thousands of grid points. The Kreiss constant is scalar and loses the geometric structure of the pseudospectrum. Crucially, pseudospectral methods operate on $\mathbb{C}$ and do not naturally integrate with statistical estimation, regularisation, or Bayesian uncertainty quantification. Section 8 shows how the pseudospectral abscissa can be bounded from above using the information-geometric commutator norm, providing a computationally efficient alternative.

3.5. Stein Estimation and Covariance Geometry

In statistical applications—ecology (estimation of community matrices from time-series data), finance (estimation of drift matrices from price data), and machine learning (estimation of Jacobians from training trajectories)—the interaction matrix A is not directly observed, and must thus be estimated. The quality of the estimate depends on the choice of loss function, and the most natural loss for positive-definite precision matrices is the Stein loss [24]:

$$L_S(S_0,S_1)=\mathrm{tr}\left(S_1^{-1}{S}_0\right)-\log \det \left(S_1^{-1}{S}_0\right)-n.$$

(13)

The Stein loss has the important properties that it is invariant under inversion ($L_S(S_0,S_1)=L_S(S_1^{-1},S_0^{-1})$), is asymmetric ($L_S(S_0,S_1)\ne L_S(S_1,S_0)$ in general), and is the natural divergence for the Gaussian covariance family under a Jeffreys-style prior.

Ledoit and Wolf [25,26] showed that the sample covariance matrix $\widehat{\mathsf{\Sigma }}$ is an inadmissible estimator of the true covariance $\mathsf{\Sigma }$ under the Stein loss for $n\ge 3$, and that nonlinear (oracle) shrinkage towards the identity substantially improves estimation accuracy in the large-n regime. This observation has motivated a large body of literature on covariance regularisation for financial networks [27] and ecological networks [28].

• Limitations. The Stein-loss framework operates separately from stability analysis: it optimises estimation quality without any direct constraint that the estimated matrix $\widehat{A}$ should be stable (i.e., $\omega \left(\widehat{A}\right)<0$). Stability constraints can be imposed post hoc (projecting $\widehat{A}$ onto the stable set), but this projection has no information-geometric interpretation and may introduce large distortions. Section 7 of this paper shows that the Stein loss is precisely the Bregman divergence of the dually flat structure on $({\mathcal{M}}_n^{+},g)$, so that the information-geometric framework provides a principled unified treatment of stability and estimation via the same geometric object.

3.6. Limitations of Existing Approaches and the Case for a Geometric Framework

The methods surveyed above are powerful in their respective domains but share a common structural limitation: they treat stability as a binary predicate (stable or not) or a scalar threshold ($\alpha <0$ or $\omega <0$), without endowing the space of interaction matrices with a geometry that would make distances, geodesics, regularisation, and estimation all coherent concepts simultaneously.

Concretely, three questions are not answered by any of the methods above. First: Given two stable interaction matrices $A_0$ and $A_1$, what is the natural interpolation path between them that stays within the stable set? Neither Lyapunov theory, numerical abscissa, nor pseudospectral methods provide a canonical answer. Second: Given an unstable estimated matrix $\widehat{A}$, what is the nearest stable matrix in a statistically motivated sense? Projecting onto $\{\omega <0\}$ ignores the curvature of the stable set. Third: How should the stability margin $-\omega \left(A\right)$ and the transient amplification risk $\mathcal{C}\left(A\right)$ be jointly monitored, and what is the correct trade-off between them?

The information-geometric framework developed in the remainder of this paper answers all three questions: geodesics in $({\mathcal{M}}_n^{+},g)$ provide the canonical interpolation path (Section 6); the Riemannian gradient descent algorithm of Section 9.2 gives the metrically nearest stable matrix in the Fisher–Rao sense; and the sectional-curvature theorem (Theorem 3) and the transient-growth bound (Theorem 7) together provide the joint certificate.

4. The Information Manifold of Asymmetric Interaction Matrices

4.1. The Manifold ${\mathcal{M}}_n^{+}$ and Its Tangent Structure

Let ${\mathcal{M}}_n^{+}≔\{A\in {\mathbb{R}}_{\mathrm{c}}^{n\times n}:S\left(A\right)\prec 0\}$. Since $S:{\mathbb{R}}_{\mathrm{c}}^{n\times n}\to {\mathrm{Sym}}_n$ is a continuous linear map and the cone of negative-definite matrices is open in ${\mathrm{Sym}}_n$, the set ${\mathcal{M}}_n^{+}$ is open in ${\mathbb{R}}_{\mathrm{c}}^{n\times n}$. It is therefore a smooth manifold of dimension $n^2$, and $T_A{\mathcal{M}}_n^{+}\cong {\mathbb{R}}_{\mathrm{c}}^{n\times n}$ for every $A\in {\mathcal{M}}_n^{+}$. Writing ${\mathrm{SPD}}_n$ for the cone of symmetric positive-definite matrices, the manifold ${\mathcal{M}}_n^{+}$ decomposes as the direct product ${\mathcal{M}}_n^{+}\cong {\mathrm{SPD}}_n\times {\mathrm{Skw}}_n$ via the map $A\mapsto (-S(A),K(A\left)\right)$, which is a diffeomorphism with inverse $(P,K)\mapsto -P+K$. The positive-definite representative $P=-S\left(A\right)\succ 0$ is precisely the precision matrix of the Gaussian $\mathcal{N}(0,P^{-1})=\mathcal{N}(0,-S{\left(A\right)}^{-1})$ associated with A, and all Fisher–Rao geometric quantities are expressed through $-S\left(A\right)$.

Under this identification, the tangent space at $A=S+K$ decomposes orthogonally as $T_A{\mathcal{M}}_n^{+}={\mathrm{Sym}}_n{\oplus }_F{\mathrm{Skw}}_n$, where the subscript F denotes orthogonality in the Frobenius sense. An arbitrary tangent vector $\delta A\in T_A{\mathcal{M}}_n^{+}$ has the symmetric part $\delta S=S\left(\delta A\right)=(\delta A+{\delta A}^{\mathsf{T}})/2\in {\mathrm{Sym}}_n$ and skew-symmetric part $\delta K=K\left(\delta A\right)=(\delta A-{\delta A}^{\mathsf{T}})/2\in {\mathrm{Skw}}_n$.

4.2. The Fisher–Rao Pullback and Its Degeneracy

The smooth map $P:{\mathcal{M}}_n^{+}\to {\mathrm{SPD}}_n$, $A\mapsto -(A+A^{\mathsf{T}})/2=-S\left(A\right)$ induces a pullback of the Fisher–Rao metric (6) to ${\mathcal{M}}_n^{+}$:

$${\tilde{g}}_A≔(\delta A,\delta A^{\prime })g_{P\left(A\right)}^F\left(\delta P,\delta P^{\prime }\right)=\frac{1}{2}\mathrm{tr}\left(P{\left(A\right)}^{-1}\delta PP{\left(A\right)}^{-1}\delta P^{\prime }\right)=\frac{1}{2}\mathrm{tr}\left(S{\left(A\right)}^{-1}\delta SS{\left(A\right)}^{-1}\delta S^{\prime }\right),$$

(14)

where $P\left(A\right)=-S\left(A\right)\succ 0$, $\delta P=-S\left(\delta A\right)$, $\delta P^{\prime }=-S\left(\delta A^{\prime }\right)$, $\delta S=S\left(\delta A\right)$, and $\delta S^{\prime }=S\left(\delta A^{\prime }\right)$; the final equality holds because the two sign changes in $\delta P=-\delta S$ cancel. The following proposition characterises the degeneracy of this pullback.

Proposition 4

(Degeneracy of the Pullback Metric). The bilinear form ${\tilde{g}}_A$ defined in (14) is positive semi-definite on $T_A{\mathcal{M}}_n^{+}$ and degenerate precisely along the subspace ${\mathrm{Skw}}_n\subset T_A{\mathcal{M}}_n^{+}$, that is, ${\tilde{g}}_A(\delta A,\delta A^{\prime })=0$ for all $\delta A^{\prime }\in T_A{\mathcal{M}}_n^{+}$ if and only if $\delta A\in {\mathrm{Skw}}_n$.

Proof. 

Positive semi-definiteness follows because $g_{P\left(A\right)}^F$ is positive-definite on ${\mathrm{Sym}}_n$ and ${\tilde{g}}_A(\delta A,\delta A)=g_{P\left(A\right)}^F(\delta P,\delta P)\ge 0$. For the degeneracy claim, observe that ${\tilde{g}}_A(\delta A,\delta A^{\prime })=0$ for all $\delta A^{\prime }$ forces $g_{P\left(A\right)}^F(\delta P,·)\equiv 0$, which (since $g_{S\left(A\right)}^F$ is non-degenerate) implies $\delta S=0$, i.e., $\delta A=\delta K\in {\mathrm{Skw}}_n$. Conversely, if $\delta A\in {\mathrm{Skw}}_n$, then $\delta S=0$ and ${\tilde{g}}_A(\delta A,\delta A^{\prime })=0$ for all $\delta A^{\prime }$. □

Proposition 4 shows that $\tilde{g}$ does not see the skew-symmetric degrees of freedom. This is statistically natural: two interaction matrices A and $A+\delta K$ (differing only by a skew-symmetric perturbation) induce identical Gaussian distributions and hence are statistically indistinguishable. However, for the study of dynamical systems, the skew-symmetric part is essential, as it encodes the anti-symmetric (circulation) component of the interaction. We therefore augment the metric.

4.3. The Augmented Metric

Definition 6

(Augmented Information Metric). For a parameter $c>0$, the augmented information metric on ${\mathcal{M}}_n^{+}$ is

$$g_A(\delta A,\delta A^{\prime })≔\underset{\mathrm{Fisher}–\mathrm{Rao}\mathrm{part}}{\underbrace{\frac{1}{2}\mathrm{tr}\left(S^{-1}\delta S{S}^{-1}\delta S^{\prime }\right)}}+\underset{\mathrm{Frobenius}\mathrm{part}}{\underbrace{c\mathrm{tr}\left(\delta K{\delta K}^{\mathsf{T}}\right)}},$$

(15)

where $S=S\left(A\right)$, $\delta S=S\left(\delta A\right)$, $\delta K=K\left(\delta A\right)$, and similarly for primed quantities.

The metric g is the direct sum of $g^F$ and $c·g^{\mathrm{Frob}}$ on the product decomposition ${\mathcal{M}}_n^{+}\cong {\mathrm{SPD}}_n\times {\mathrm{Skw}}_n$. Being the sum of two positive-definite bilinear forms, it is itself positive-definite on all of $T_A{\mathcal{M}}_n^{+}$, making $({\mathcal{M}}_n^{+},g)$ a genuine Riemannian manifold. The constant c is a free parameter that controls the relative weighting between “informational” and “circulatory” degrees of freedom; for concreteness, we set $c=1$ in all that follows, noting that the qualitative geometric results are c-independent.

The metric (15) decomposes the squared norm of a tangent vector as

$$g_A(\delta A,\delta A)=\frac{1}{2}\mathrm{tr}\left(S^{-1}\delta S{S}^{-1}\delta S\right)+\mathrm{tr}\left(\delta K{\delta K}^{\mathsf{T}}\right).$$

(16)

When $\delta A=\delta S\in {\mathrm{Sym}}_n$ (purely symmetric perturbation), the metric reduces to the Fisher–Rao metric on ${\mathrm{SPD}}_n$. When $\delta A=\delta K\in {\mathrm{Skw}}_n$ (purely skew-symmetric), the metric reduces to the standard Frobenius metric on ${\mathrm{Skw}}_n$. The two components are g-orthogonal by construction.

4.4. Curvature of $({\mathcal{M}}_n^{+},g)$

The Riemannian curvature of the product manifold $({\mathcal{M}}_n^{+},g)\cong ({\mathrm{SPD}}_n,g^F)\times ({\mathrm{Skw}}_n,g^{\mathrm{Frob}})$ is the direct sum of the curvatures of its factors. The Euclidean factor $({\mathrm{Skw}}_n,g^{\mathrm{Frob}})$ has zero curvature. The Fisher–Rao factor $({\mathrm{SPD}}_n,g^F)$ has well-known non-positive sectional curvature. We now compute the curvature of $({\mathcal{M}}_n^{+},g)$ explicitly.

Theorem 3

(Sectional Curvature of $({\mathcal{M}}_n^{+},g)$). Let $A=S+K\in {\mathcal{M}}_n^{+}$ and let $e_S\in {\mathrm{Sym}}_n$ and $e_K\in {\mathrm{Skw}}_n$ be unit tangent vectors at A (i.e., $g_A(e_S,e_S)=g_A(e_K,e_K)=1$). The sectional curvature of $({\mathcal{M}}_n^{+},g)$ in the $(e_S,e_K)$-plane satisfies

$$\sec (e_S,e_K)={\kappa }_{\mathrm{FR}}(S^{1/2}{e}_S{S}^{1/2},0)\le 0,$$

(17)

where ${\kappa }_{\mathrm{FR}}$ denotes the sectional curvature of $({\mathrm{SPD}}_n,g^F)$. For any two unit vectors $e_1,e_2\in {\mathrm{Sym}}_n$ at $S=I$ (the identity),

$${\kappa }_{\mathrm{FR}}(e_1,e_2)=-\frac{1}{4}{\displaystyle \frac{{∥[e_1,e_2]∥}_F^2}{{∥e_1∥}_F^2{∥e_2∥}_F^2-{\langle e_1,e_2\rangle }_F^2}}.$$

(18)

Consequently, the sectional curvature $\sec (e_S,e_K)$ vanishes if and only if $[e_S,0]=0$, which holds trivially, and the curvature in the purely symmetric plane $\sec (e_1,e_2)$ for $e_1,e_2\in {\mathrm{Sym}}_n$ vanishes if and only if $[e_1,e_2]=0$.

Proof. 

Since $({\mathcal{M}}_n^{+},g)$ is a Riemannian product, cross-sectional planes containing one symmetric and one skew factor have curvature equal to zero by the product structure (mixed cross-curvature in a product manifold vanishes identically). For planes within the ${\mathrm{SPD}}_n$ factor, the sectional curvature of $({\mathrm{SPD}}_n,g^F)$ at the identity $S=I$ is given by (18), which follows from the O’Neill formula for submersions [6] or directly from the Lie group structure of $GL\left(n\right)$ [16]. The formula at a general point $S\in {\mathrm{SPD}}_n$ is obtained by conjugation: the map $X\mapsto S^{-1/2}X{S}^{-1/2}$ is an isometry of $({\mathrm{SPD}}_n,g^F)$, and formula (18) applies after this normalisation. The vanishing condition follows immediately from the formula. A self-contained derivation of Equation (18) from the O’Neill submersion formula is given in Appendix A. □

Remark 2

(Normality and Curvature). While the sectional curvature in the mixed $(S,K)$ plane vanishes by the product structure, the normality of A manifests in the curvature within the ${\mathrm{SPD}}_n$ factor. Specifically, the curvature in the direction of S at a point $A=S+K$ is related to the commutator $[S,K]$ through the second variation in the Fisher–Rao distance from S to nearby symmetric-part matrices. This connection is made precise in Theorem 4 below.

We now establish the central geometric characterisation of normality.

Theorem 4

(Normality as a Flat Direction). Let $A\in {\mathcal{M}}_n^{+}$ with $A=S+K$. The path $\gamma :(-\epsilon ,\epsilon )\to {\mathcal{M}}_n^{+}$ defined by $\gamma \left(t\right)=(S+t{K}^{\prime })+K$ (varying the symmetric part in direction $K^{\prime }\in {\mathrm{Sym}}_n$) has zero geodesic curvature in $({\mathrm{SPD}}_n,g^F)$ if and only if $[S,K^{\prime }]=0$. More generally, the Hessian of the squared Fisher–Rao distance $\varphi \left(S^{\prime }\right)≔{\displaystyle \frac{1}{2}}{\left(d^F(S,S^{\prime })\right)}^2$ at $S^{\prime }=S$ satisfies

$${\mathrm{Hess}\varphi |}_{S^{\prime }=S}(U,V)=g_S^F(U,V),$$

(19)

and the third-order curvature correction involves the commutator $[S^{-1/2}U{S}^{-1/2},S^{-1/2}V{S}^{-1/2}]$ [16].

Theorem 5

(Normality as a Geometric Condition). A matrix $A=S+K\in {\mathcal{M}}_n^{+}$ is normal if and only if $[S,K]=0$, and this condition is equivalent to the following: the pair $(S,K)$ lies in a maximal Abelian subalgebra of $({\mathrm{Sym}}_n\oplus {\mathrm{Skw}}_n,[·,·])$, or equivalently, A is simultaneously diagonalisable with S and $-A^{\mathsf{T}}=-S+K$ via an orthogonal similarity.

Proof. 

The equivalence $A{A}^{\mathsf{T}}=A^{\mathsf{T}}A\iff [S,K]=0$ is a standard algebraic identity: expanding $A{A}^{\mathsf{T}}=(S+K)(S-K)=S^2-SK+KS-K^2$ and $A^{\mathsf{T}}A=(S-K)(S+K)=S^2+SK-KS-K^2$, so $A{A}^{\mathsf{T}}-A^{\mathsf{T}}A=-2SK+2KS=-2[S,K]$. Thus, $A{A}^{\mathsf{T}}=A^{\mathsf{T}}A$ iff $[S,K]=0$. The second equivalence (maximal Abelian subalgebra) follows from the commutativity condition; the third (orthogonal simultaneous diagonalisation) follows because normal real matrices are orthogonally block-diagonalisable [9]. □

4.5. The Information-Geometric Stability Boundary

For $A\in {\mathcal{M}}_n^{+}$, the positive-definite representative is $P\left(A\right)≔-S\left(A\right)\succ 0$, and the numerical stability margin is $\left|\omega \left(A\right)\right|={\lambda }_{\min }\left(P\left(A\right)\right)>0$. The instability boundary of ${\mathcal{M}}_n^{+}$ is the set $\partial {\mathcal{M}}_n^{+}=\{A\in {\mathbb{R}}_{\mathrm{c}}^{n\times n}:{\lambda }_{\min }\left(P\left(A\right)\right)=0\}=\{A:\omega \left(A\right)=0\}$, where the negative-definiteness of the symmetric part breaks down. Under the isomorphism ${\mathcal{M}}_n^{+}\cong {\mathrm{SPD}}_n\times {\mathrm{Skw}}_n$, $A\mapsto \left(P\right(A),K(A\left)\right)$, this corresponds to approaching the boundary $\partial {\mathrm{SPD}}_n=\{X\in {\mathrm{Sym}}_n:{\lambda }_{\min }\left(X\right)=0\}$ of the positive-definite cone.

Lemma 1

(Log-Lipschitz Continuity of the Minimal Eigenvalue). For all $X,Y\in {\mathrm{SPD}}_n$,

$$|\log {\lambda }_{\min }\left(X\right)-\log {\lambda }_{\min }\left(Y\right)|\le {∥\log \left(Y^{-1/2}X{Y}^{-1/2}\right)∥}_2\le d^F(X,Y).$$

Proof. 

Let $d_T≔{∥\log \left(Y^{-1/2}X{Y}^{-1/2}\right)∥}_2$ (the Thompson metric). By definition of $d_T$, the eigenvalues of $Y^{-1/2}X{Y}^{-1/2}$ lie in $[e^{-d_T},e^{d_T}]$; hence, $e^{-d_T}Y⪯X⪯e^{d_T}Y$ in the Loewner order. Monotonicity of ${\lambda }_{\min }$ under the Loewner order (Weyl) gives $e^{-d_T}{\lambda }_{\min }\left(Y\right)\le {\lambda }_{\min }\left(X\right)\le e^{d_T}{\lambda }_{\min }\left(Y\right)$, i.e., $|\log {\lambda }_{\min }\left(X\right)-\log {\lambda }_{\min }\left(Y\right)|\le d_T$. Finally, $d_T\le d^F(X,Y)$ because the spectral norm of $\log \left(Y^{-1/2}X{Y}^{-1/2}\right)$ is dominated by its Frobenius norm, which is precisely $d^F(X,Y)$ by (7). □

Proposition 5

(The Stability Margin and the Boundary at Infinity). Let $A\in {\mathcal{M}}_n^{+}$ with positive-definite representative $P=P\left(A\right)=-S\left(A\right)\succ 0$ and eigenvalues $0<{\mu }_1\le \cdots \le {\mu }_n$ of P, so ${\mu }_1={\lambda }_{\min }\left(P\right)=\left|\omega \left(A\right)\right|$. Then:

(i)

(Boundary at infinite distance.) $d^F(P,\partial {\mathrm{SPD}}_n)=+\infty$: Every piecewise-smooth path in ${\mathrm{SPD}}_n$ from P to a point of $\partial {\mathrm{SPD}}_n$ has infinite Fisher–Rao length. The instability boundary is therefore unreachable in finite Fisher–Rao distance, which is the precise quantitative content of the Hadamard completeness of $({\mathrm{SPD}}_n,g^F)$ in this context.

(ii)

(The signed stability margin.) Define the Fisher–Rao stability margin

$$m^F\left(A\right)≔\log {\lambda }_{\min }\left(P\right)=\log \left|\omega \left(A\right)\right|\in \mathbb{R}.$$

(20)

Then, $|m^F\left(A\right)|=d^F(P,\widehat{P})$, where $\widehat{P}≔P+(1-{\mu }_1)v_1{v}_1^{\top }$ (with $v_1$ as a unit eigenvector for ${\mu }_1$) is the matrix obtained from P by renormalising its most fragile mode to unit precision; $m^F\left(A\right)\to -\infty$ as $\omega \left(A\right)\to 0^{-}$ (approach to instability), $m^F\left(A\right)\to +\infty$ as ${\lambda }_{\min }\left(P\right)\to \infty$ (deep stability), and $m^F\left(A\right)=0$ exactly at ${\lambda }_{\min }\left(P\right)=1$.

(iii)

(Euclidean comparison.) In the spectral norm, the Euclidean distance from P to the boundary is finite and equals the numerical margin as follows: ${\mathrm{dist}}_{{∥·∥}_2}(P,\partial {\mathrm{SPD}}_n)={\lambda }_{\min }\left(P\right)=\left|\omega \left(A\right)\right|$.

Proof. 

(i) Let $\gamma :[0,1)\to {\mathrm{SPD}}_n$ be piecewise smooth with $\gamma \left(0\right)=P$ and $\gamma \left(t\right)\to X_0\in \partial {\mathrm{SPD}}_n$ as $t\to 1^{-}$. Then, ${\lambda }_{\min }\left(\gamma \left(t\right)\right)\to {\lambda }_{\min }\left(X_0\right)=0$, so $\log {\lambda }_{\min }\left(\gamma \left(t\right)\right)\to -\infty$. By Lemma 1, for every t,

$$\mathrm{Length}\left(\gamma {|}_{[0,t]}\right)\ge d^F(P,\gamma \left(t\right))\ge \left|\log {\lambda }_{\min }\left(\gamma \left(t\right)\right)-\log {\lambda }_{\min }\left(P\right)\right|⟶+\infty ,$$

so the total length is infinite. Since $\gamma$ is arbitrary, $d^F(P,\partial {\mathrm{SPD}}_n)=+\infty$.

(ii) Work in the eigenbasis of P, so $P=\mathrm{diag}({\mu }_1,\dots ,{\mu }_n)$ and $\widehat{P}=\mathrm{diag}(1,{\mu }_2,\dots ,{\mu }_n)$. The relative matrix $P^{-1/2}\widehat{P}{P}^{-1/2}=\mathrm{diag}(1/{\mu }_1,1,\dots ,1)$ has eigenvalues $(1/{\mu }_1,1,\dots ,1)$, so by the distance formula (7), $d^F(P,\widehat{P})={\left({\log }^2(1/{\mu }_1)\right)}^{1/2}=|\log {\mu }_1|=|m^F\left(A\right)|$. The limits and the zero at ${\mu }_1=1$ are immediate from $m^F=\log {\mu }_1$.

(iii) For any $X\in \partial {\mathrm{SPD}}_n$, Weyl’s inequality gives $|{\lambda }_{\min }\left(P\right)-{\lambda }_{\min }\left(X\right)|\le {∥P-X∥}_2$, and ${\lambda }_{\min }\left(X\right)=0$, so ${∥P-X∥}_2\ge {\lambda }_{\min }\left(P\right)$. The bound is attained by $X=P-{\mu }_1{v}_1{v}_1^{\top }\in \partial {\mathrm{SPD}}_n$, for which ${∥P-X∥}_2={\mu }_1$. □

Proposition 5 corrects an imprecision in earlier versions of this work, where the finite quantity $|\log {\lambda }_{\min }\left(P\right)|$ was described as the “distance to the instability boundary.” The boundary is in fact at infinite Fisher–Rao distance—in part (i), which is the rigorous answer to how completeness manifests itself, the stable cone is geodesically complete, so no trajectory of finite information-geometric length can exit it. The finite quantity is the signed margin $m^F$ of part (ii): its magnitude is the exact geodesic cost of renormalising the most fragile mode to unit precision, its sign separates the fragile regime (${\lambda }_{\min }\left(P\right)<1$, $m^F<0$) from the robust regime (${\lambda }_{\min }\left(P\right)>1$, $m^F>0$), and $m^F\to -\infty$ precisely as the system approaches instability. Throughout the remainder of this paper, statements previously phrased as “Fisher–Rao distance to the boundary” refer to this margin. The two-sided behaviour is now transparent rather than puzzling: deep stability ($m^F\to +\infty$) and imminent instability ($m^F\to -\infty$) sit at opposite ends of a single signed scale, with the Euclidean margin of part (iii) supplying the complementary linear measure ${\lambda }_{\min }\left(P\right)=\left|\omega \left(A\right)\right|$. That ${\lambda }_{\min }\left(P\right)\to +\infty$ is attainable by finite matrices is elementary: for the family

$$A_k=-k{I}_n,k>0,$$

(21)

one has $-S\left(A_k\right)=k{I}_n$ with ${\lambda }_{\min }(-S\left(A_k\right))=k$, so $m^F\left(A_k\right)=\log k\to +\infty$ as $k\to \infty$; no infinite dimension is needed since the eigenvalues of a finite symmetric matrix are unbounded above.

A worked boundary crossing. To see the boundary geometry concretely, consider the $2\times 2$ family

$$A_{ϵ}=\left(\begin{array}{cc}-ϵ& 1\\ -1& -ϵ\end{array}\right),ϵ>0.$$

(22)

Its symmetric part is $S\left(A_{ϵ}\right)=-ϵI_2$ and its skew part is $K\left(A_{ϵ}\right)=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$. The eigenvalues of $A_{ϵ}$ are $-ϵ\pm i$, so $\alpha \left(A_{ϵ}\right)=-ϵ$, while $\omega \left(A_{ϵ}\right)={\lambda }_{\max }\left(S\left(A_{ϵ}\right)\right)=-ϵ$ as well; the matrix is normal (indeed, $A_{ϵ}{A}_{ϵ}^{\top }=A_{ϵ}^{\top }{A}_{ϵ}=(1+{ϵ}^2)I_2$), so its spectral and numerical abscissae coincide. As $ϵ\to 0^{+}$, the symmetric part $S\left(A_{ϵ}\right)=-ϵI_2\to 0$, i.e., the representative $P=ϵI_2$ approaches $\partial {\mathrm{SPD}}_2$; the signed margin $m^F\left(A_{ϵ}\right)=\log ϵ\to -\infty$ (imminent instability) even though, by Proposition 5(i), the boundary itself remains at infinite Fisher–Rao distance; and the matrix $A_{ϵ}\to \left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$, a pure rotation generator that lies exactly on the instability boundary ($\omega =0$). This illustrates how the instability boundary is reached: not by the eigenvalues alone—which here move continuously to $\pm i$—but by the symmetric part losing definiteness.

The boundary as a stratified variety. We next formulate the boundary geometry precisely. The admissible symmetric parts form the open convex cone ${\mathrm{SPD}}_n\subset {\mathrm{Sym}}_n$. Its topological boundary is

$$\partial {\mathrm{SPD}}_n=\{X\in {\mathrm{Sym}}_n:X⪰0,\det X=0\}={⨆}_{r=0}^{n-1}{\mathcal{B}}_r,{\mathcal{B}}_r=\{X⪰0:\mathrm{rank}X=r\},$$

(23)

where each stratum ${\mathcal{B}}_r$ is a smooth manifold of dimension $rn-\left({\displaystyle \genfrac{}{}{0pt}{}{r}{2}}\right)$ [16], and the top stratum ${\mathcal{B}}_{n-1}$ (rank-deficiency one) is the part reached generically as ${\lambda }_{\min }\left(P\right)\to 0^{+}$. Thus, $\partial {\mathrm{SPD}}_n$ is a stratified hypersurface rather than a smooth submanifold, and a matrix A approaches the instability boundary exactly when its representative $P=-S\left(A\right)$ approaches ${\mathcal{B}}_{n-1}$, i.e., when a single eigenvalue of P vanishes.

The non-normality gap region. Between spectral and information-geometric stability lies the non-normality gap region

$$\mathcal{G}=\{A\in {\mathbb{R}}_{\mathrm{c}}^{n\times n}:\alpha \left(A\right)<0\le \omega \left(A\right)\},$$

(24)

whose structure we record precisely.

Proposition 6

(Structure of the Gap Region). The set $\mathcal{G}$ of (24) satisfies:

(i)

$\mathcal{G}$ contains no normal matrices; equivalently, $\mathcal{G}\cap \{A:\alpha (A)=\omega (A\left)\right\}=⌀$.

(ii)

For every $A\in \mathcal{G}$, the representative $P=-S\left(A\right)$ satisfies ${\lambda }_{\min }\left(P\right)=-\omega \left(A\right)\le 0$, so $P\notin {\mathrm{SPD}}_n$ and the distance (20) is not defined on $\mathcal{G}$; the gap region is precisely the set of spectrally stable matrices lying outside the information-geometric stability domain ${\mathcal{M}}_n^{+}$.

(iii)

$\mathcal{G}$ is non-empty for every $n\ge 2$ and is not convex.

Proof. 

(i) If A is normal, then $\alpha \left(A\right)=\omega \left(A\right)$ by Theorem 1, so the defining inequalities $\alpha \left(A\right)<0\le \omega \left(A\right)$ cannot both hold; hence, $\mathcal{G}$ contains no normal matrix. (ii) By definition, $\omega \left(A\right)={\lambda }_{\max }\left(S\left(A\right)\right)=-{\lambda }_{\min }(-S\left(A\right))=-{\lambda }_{\min }\left(P\right)$, so $\omega \left(A\right)\ge 0$ is equivalent to ${\lambda }_{\min }\left(P\right)\le 0$, i.e., $P\notin {\mathrm{SPD}}_n$; since the margin $m^F\left(A\right)=\log {\lambda }_{\min }\left(P\right)$ requires ${\lambda }_{\min }\left(P\right)>0$, it is undefined on $\mathcal{G}$, which is therefore exactly $\{A:\alpha \left(A\right)<0\}\setminus {\mathcal{M}}_n^{+}$. (iii) Non-emptiness: For $n=2$, take $A=\left(\begin{array}{cc}-1& c\\ 0& -1\end{array}\right)$ with $c>2$. Its only eigenvalue is $-1$, so $\alpha \left(A\right)=-1<0$, while $S\left(A\right)=\left(\begin{array}{cc}-1& c/2\\ c/2& -1\end{array}\right)$ has eigenvalues $-1\pm c/2$, so $\omega \left(A\right)=-1+c/2>0$; hence, $A\in \mathcal{G}$, and the construction embeds into ${\mathbb{R}}_{\mathrm{c}}^{n\times n}$ for $n>2$ by direct sum with $-I_{n-2}$. Non-convexity: With A as above and $A^{\prime }=A^{\mathsf{T}}=\left(\begin{array}{cc}-1& 0\\ c& -1\end{array}\right)\in \mathcal{G}$ (same eigenvalues and same symmetric part), the midpoint $\frac{1}{2}(A+A^{\prime })=\left(\begin{array}{cc}-1& c/2\\ c/2& -1\end{array}\right)$ is symmetric, hence normal, so by (i), it does not lie in $\mathcal{G}$. Therefore, $\mathcal{G}$ is not convex. □

On $\mathcal{G}$, a matrix is spectrally stable ($\alpha <0$), yet has $\omega \ge 0$, so it exhibits long-run decay together with finite-time transient growth ${\sup }_{t\ge 0}{∥e^{tA}∥}_2>1$; this is the regime in which the information-geometric certificate adds information beyond the spectrum, and it is the precise content summarised in Remark 3.

Remark 3

(Boundary Geometry and the Non-Normality Gap Region). Three clarifications are in order. (i) Finite-dimensional divergence. The limit ${\lambda }_{\min }\left(S\right)\to \infty$ is elementary for finite matrices: taking $P=k{I}_n$ gives ${\lambda }_{\min }\left(P\right)=k\to \infty$, and hence margin $m^F=\log k\to +\infty$ as $k\to \infty$, while $m^F\to -\infty$ as ${\lambda }_{\min }\left(P\right)\to 0^{+}$ (imminent instability); the margin vanishes only at ${\lambda }_{\min }\left(P\right)=1$. By Proposition 5(i), the boundary itself is at infinite Fisher–Rao distance in either case. (ii) The margin is computable from this proposition. Equation (20) gives the stability margin directly: it requires only the extreme eigenvalue of the symmetric part, obtained in $O\left(n^3\right)$ via a symmetric eigensolver, with no eigenvalue decomposition of the full non-symmetric matrix A. (iii) The intermediate region. The set of definite symmetric parts is an open convex cone in ${\mathrm{Sym}}_n$ whose topological boundary $\partial {\mathrm{SPD}}_n$ is the algebraic variety of singular symmetric matrices. Between spectral stability and information-geometric stability lies the non-normality gap region $\mathcal{G}=\{A:\alpha (A)<0\le \omega (A\left)\right\}$: matrices that are spectrally Hurwitz stable yet sit outside the information-geometric stable set. By Theorem 1, $\mathcal{G}$ is empty for normal matrices (where $\alpha =\omega$); for non-normal matrices, it is a connected but non-convex region in which long-run stability coexists with transient amplification, as illustrated by the matrix $A_{ϵ}$ of Remark 1.

5. Curvature-Mediated Stability Theory

5.1. The Stability Theorem

The fundamental inequality $\alpha \left(A\right)\le \omega \left(A\right)={\lambda }_{\max }\left(S\left(A\right)\right)$, combined with the information-geometric stability margin of Proposition 5, yields the following central result.

Theorem 6

(Information-Geometric Stability Certificate). Let $A\in {\mathcal{M}}_n^{+}$. The following hold.

(i) 

(Stability via $\omega$.) $\omega \left(A\right)<0$ implies $\alpha \left(A\right)<0$ (asymptotic stability). The margin $-\omega \left(A\right)=|{\lambda }_{\max }\left(S\right)|$ is a conservative stability radius, with the following information-geometric certificate:

$$m^F\left(A\right)=\log {\lambda }_{\min }(-S\left(A\right))=\log \left|\omega \left(A\right)\right|$$

(25)

where large values indicate robust stability.

(ii) 

(Transient bound.) For all $t\ge 0$:

$${∥e^{tA}∥}_2\le e^{t\omega \left(A\right)}.$$

(26)

(iii) 

(Tightness for normal matrices.) For normal A, $\alpha \left(A\right)=\omega \left(A\right)$ (Theorem 1) and the bound in (26) is tight: ${\lim }_{t\to \infty }{t}^{-1}\log {∥e^{tA}∥}_2=\alpha \left(A\right)=\omega \left(A\right)$.

Proof. 

For (i): $\omega \left(A\right)<0$ means ${\lambda }_{\max }\left(S\left(A\right)\right)<0$, so $S\left(A\right)\prec 0$. For any unit vector $\mathbf{v}$, $\mathrm{Re}\left({\mathbf{v}}^{*}\lambda \mathbf{v}\right)={\mathbf{v}}^{*}S\left(A\right)\mathbf{v}\le {\lambda }_{\max }\left(S\left(A\right)\right)<0$ for any eigenvalue $\lambda$ with eigenvector $\mathbf{v}$, so $\alpha \left(A\right)<0$. For (ii): The bound ${∥e^{tA}∥}_2\le e^{t\omega \left(A\right)}$ follows from the identity ${\displaystyle \frac{d}{dt}}{∥e^{tA}\mathbf{v}∥}^2=2{\mathbf{v}}^T{e}^{t{A}^T}\left(S\left(A\right)\right)e^{tA}\mathbf{v}\le 2\omega \left(A\right){∥e^{tA}\mathbf{v}∥}^2$, integrating via Grönwall’s inequality [12]. For (iii): The equality $\alpha =\omega$ for normal matrices (Theorem 1) and the sub-multiplicativity of the matrix exponential give both the upper and lower bounds. □

5.2. The Non-Normality Defect and Curvature

The difference $\omega \left(A\right)-\alpha \left(A\right)\ge 0$ measures the conservatism of the numerical abscissa as a stability indicator. We now relate this quantity to the information-geometric curvature.

Definition 7

(Non-Normality Defect). The non-normality defect of $A\in {\mathbb{R}}_{\mathrm{c}}^{n\times n}$ is $\mathcal{N}\left(A\right)≔{∥A∥}_F^2-{\sum }_{i=1}^n{\left|{\lambda }_i\left(A\right)\right|}^2\ge 0$, which vanishes if and only if A is normal. The commutator norm is $\mathcal{C}\left(A\right)≔{∥\left[S\right(A),K(A\left)\right]∥}_F$.

Proposition 7

(Commutator and Non-Normality). For any $A\in {\mathbb{R}}_{\mathrm{c}}^{n\times n}$, $\mathcal{C}\left(A\right)=0$ if and only if A is normal. The Frobenius norm of the commutator satisfies $\mathcal{C}\left(A\right)\le 2{∥S\left(A\right)∥}_F{∥K\left(A\right)∥}_F$.

Proof. 

The equivalence $\mathcal{C}\left(A\right)=0\iff A$ is normal, following from Theorem 5. For the bound, ${∥[S,K]∥}_F={∥SK-KS∥}_F\le {∥SK∥}_F+{∥KS∥}_F\le 2{∥S∥}_F{∥K∥}_F$ by sub-multiplicativity of the Frobenius norm [9]. □

Theorem 7

(Curvature Bound on Transient Growth). Let $A=S+K\in {\mathcal{M}}_n^{+}$ with $\omega \left(A\right)<0$. The maximum transient growth factor satisfies

$$\underset{t\ge 0}{\sup }{∥e^{tA}∥}_2\le \exp \left({\displaystyle \frac{\mathcal{C}{\left(A\right)}^2}{8\left|\omega \left(A\right)\right|{∥S\left(A\right)∥}_F^2}}\right),$$

(27)

where $\mathcal{C}\left(A\right)={∥\left[S\right(A),K(A\left)\right]∥}_F$ is the commutator norm. When A is normal, $\mathcal{C}\left(A\right)=0$ and the bound equals 1 (no transient growth).

Proof. 

The maximum of ${∥e^{tA}∥}_2$ over $t\ge 0$ can be bounded using the Kreiss matrix theorem and the pseudospectral analysis [20]. However, a direct information-geometric bound proceeds as follows. By the Lyapunov inequality, if $S\left(A\right)=-Q$ for some $Q\succ 0$, then for any $t\ge 0$, ${\displaystyle \frac{d}{dt}}{∥e^{tA}\mathbf{v}∥}^2=-2{\mathbf{v}}^T{e}^{t{A}^T}Q{e}^{tA}\mathbf{v}$. The non-normal component introduces a correction: writing $e^{tA}=e^{tS}{e}^{tK}+O\left(t^2[S,K]\right)$ via the Baker–Campbell–Hausdorff formula [20], the leading transient amplification from the commutator $[S,K]$ satisfies $\log {\sup }_t{∥e^{tA}∥}_2\le {\displaystyle \frac{{∥[S,K]∥}_F^2}{8\left|\omega \left(A\right)\right|{∥S∥}_F^2}}$, which gives (27). The Baker–Campbell–Hausdorff estimate is carried out in full in Appendix B. □

5.3. The Lyapunov–Fisher Correspondence

The classical Lyapunov stability theory characterises stability via the existence of a Lyapunov function. The information-geometric framework provides a canonical choice.

Theorem 8

(Lyapunov–Fisher Correspondence). Let $A\in {\mathcal{M}}_n^{+}$ with $\omega \left(A\right)<0$, and set $P≔-S\left(A\right)\succ 0$. Then:

(i) 

The quadratic form $V\left(\mathbf{x}\right)={\mathbf{x}}^T\mathbf{x}$ is a strict Lyapunov function for $\dot{\mathbf{x}}=A\mathbf{x}$, with $\dot{V}\left(\mathbf{x}\right)\le 2\omega \left(A\right){∥\mathbf{x}∥}^2<0$ for $\mathbf{x}\ne 0$.

(ii) 

The precision matrix $P=-S\left(A\right)$ is the Fisher–Rao base point of A, and the decay rate of V is controlled by the stability margin $\left|\omega \left(A\right)\right|={\lambda }_{\min }\left(P\right)$: explicitly, $\dot{V}\le -2{\lambda }_{\min }\left(P\right){∥\mathbf{x}∥}^2$.

(iii) 

The Fisher–Rao stability margin $m^F\left(A\right)=\log {\lambda }_{\min }\left(P\right)$ is monotonically related to the largest ball centred at A in $({\mathcal{M}}_n^{+},g)$ that lies entirely within the stable region $\{\omega <0\}$; the boundary $\{\omega =0\}$ itself lies at infinite Fisher–Rao distance (Proposition 5(i)).

Proof. 

For (i), with $V\left(\mathbf{x}\right)={∥\mathbf{x}∥}^2={\mathbf{x}}^T\mathbf{x}$ (the Lyapunov function associated with $P_{\mathrm{Lyap}}=I$),

$$\dot{V}={\mathbf{x}}^T(A+A^{\mathsf{T}})\mathbf{x}=2{\mathbf{x}}^{T}S\left(A\right)\mathbf{x}\le 2{\lambda }_{\max }\left(S\left(A\right)\right){∥\mathbf{x}∥}^2=2\omega \left(A\right){∥\mathbf{x}∥}^2<0,$$

using $\omega \left(A\right)={\lambda }_{\max }\left(S\left(A\right)\right)<0$. This proves (i). Part (ii) is the same inequality written through $P=-S\left(A\right)\succ 0$, for which ${\lambda }_{\max }\left(S\left(A\right)\right)=-{\lambda }_{\min }\left(P\right)$, giving $\dot{V}\le -2{\lambda }_{\min }\left(P\right){∥\mathbf{x}∥}^2$. Part (iii) follows from Proposition 5: the margin $m^F\left(A\right)=\log {\lambda }_{\min }\left(P\right)$ increases monotonically as the smallest eigenvalue of P moves away from the singular boundary $\{{\lambda }_{\min }\left(P\right)=0\}=\{\omega \left(A\right)=0\}$, which is exactly the radius of stability in the numerical-abscissa sense [5,6]. □

6. Geodesics, Exponential Map, and Parallel Transport on ${\mathcal{M}}_{\mathit{n}}^{\mathbf{+}}$

6.1. Geodesic Equations

Since $({\mathcal{M}}_n^{+},g)$ is a Riemannian product of $({\mathrm{SPD}}_n,g^F)$ and $({\mathrm{Skw}}_n,g^{\mathrm{Frob}})$, the geodesic equations decouple. A smooth curve $\gamma \left(t\right)=S\left(t\right)+K\left(t\right)\in {\mathcal{M}}_n^{+}$ is a geodesic of $({\mathcal{M}}_n^{+},g)$ if and only if $S\left(t\right)$ is a geodesic of $({\mathrm{SPD}}_n,g^F)$ and $K\left(t\right)$ is a geodesic of $({\mathrm{Skw}}_n,g^{\mathrm{Frob}})$.

Proposition 8

(Geodesics of $({\mathcal{M}}_n^{+},g)$). The geodesic emanating from $A_0=S_0+K_0\in {\mathcal{M}}_n^{+}$ with initial velocity $\dot{\gamma }\left(0\right)={\dot{S}}_0+{\dot{K}}_0$ (where ${\dot{S}}_0\in {\mathrm{Sym}}_n$, ${\dot{K}}_0\in {\mathrm{Skw}}_n$) is

$$\gamma \left(t\right)=S_0^{1/2}\exp \left(t{S}_0^{-1/2}{\dot{S}}_0{S}_0^{-1/2}\right)S_0^{1/2}+K_0+t{\dot{K}}_0,$$

(28)

where $\exp (·)$ denotes the matrix exponential. The symmetric component is the Fisher–Rao geodesic in ${\mathrm{SPD}}_n$; the skew-symmetric component is an affine (Euclidean) path in ${\mathrm{Skw}}_n$.

Proof. 

The geodesic equation for $({\mathrm{SPD}}_n,g^F)$ at $S_0$ with initial velocity ${\dot{S}}_0$ is the Christoffel-symbol equation, whose solution in the Lie-group representation $S\left(t\right)=S_0^{1/2}M\left(t\right)S_0^{1/2}$ with $M\left(0\right)=I$ reduces to $\ddot{M}+\dot{M}{M}^{-1}\dot{M}=0$; the solution is $M\left(t\right)=\exp \left(t{S}_0^{-1/2}{\dot{S}}_0{S}_0^{-1/2}\right)$, giving the first term of (28) [16]. For the flat Euclidean component $({\mathrm{Skw}}_n,g^{\mathrm{Frob}})$, the Christoffel symbols vanish and the geodesics are straight lines. □

6.2. Geodesic Distance and the Log Map

From Proposition 8, the geodesic distance between two interaction matrices $A_0,A_1\in {\mathcal{M}}_n^{+}$ is

$$d_g{(A_0,A_1)}^2=d^F{(S_0,S_1)}^2+{∥K_0-K_1∥}_F^2,$$

(29)

where $d^F(S_0,S_1)={\left({\sum }_i{\log }^2{\mu }_i\right)}^{1/2}$ with ${\mu }_i$ the eigenvalues of $S_0^{-1/2}{S}_1{S}_0^{-1/2}$, and ${∥K_0-K_1∥}_F$ is the Frobenius norm of the skew-symmetric difference.

The logarithmic map at $A_0$ is

$${\log }_{A_0}\left(A_1\right)=S_0^{1/2}\log \left(S_0^{-1/2}{S}_1{S}_0^{-1/2}\right)S_0^{1/2}+(K_1-K_0),$$

(30)

and the exponential map at $A_0=S_0+K_0$ applied to a tangent vector $V=V_S+V_K$ (with $V_S\in {\mathrm{Sym}}_n$, $V_K\in {\mathrm{Skw}}_n$) is

$${\exp }_{A_0}\left(V\right)=S_0^{1/2}\exp \left(S_0^{-1/2}{V}_S{S}_0^{-1/2}\right)S_0^{1/2}+K_0+V_K.$$

(31)

These maps are globally defined on ${\mathcal{M}}_n^{+}$ because $({\mathrm{SPD}}_n,g^F)$ is a Hadamard manifold (complete, simply connected, non-positive curvature) and ${\mathrm{Skw}}_n$ is a Euclidean space. Consequently, $({\mathcal{M}}_n^{+},g)$ is itself a Hadamard manifold.

Corollary 1

(Global Convexity of $({\mathcal{M}}_n^{+},g)$). The Riemannian manifold $({\mathcal{M}}_n^{+},g)$ is a Hadamard manifold: it is complete, simply connected, and has non-positive sectional curvature everywhere. As a consequence, there is a unique geodesic between any two points $A_0,A_1\in {\mathcal{M}}_n^{+}$, and all geodesic balls are convex.

Proof. 

Completeness of $({\mathrm{SPD}}_n,g^F)$ is standard [16]; completeness of $({\mathrm{Skw}}_n,g^{\mathrm{Frob}})$ is trivial. The Cartan–Hadamard theorem then applies to the product. □

6.3. Parallel Transport

Parallel transport along a geodesic $\gamma \left(t\right)$ in $({\mathcal{M}}_n^{+},g)$ also decouples into a symmetric and skew-symmetric component. For a tangent vector $V=V_S+V_K$ at $A_0$, its parallel transport to $\gamma \left(t\right)=S\left(t\right)+K\left(t\right)$ is

$${\mathcal{P}}_t\left(V\right)={\left(S{\left(t\right)}^{1/2}{S}_0^{-1/2}\right)}^{1/2}{V}_S{\left(S_0^{-1/2}S{\left(t\right)}^{1/2}\right)}^{1/2}+V_K.$$

(32)

The skew-symmetric part is transported trivially (constant in the Euclidean factor), while the symmetric part rotates to track the evolving metric structure. This formula is used in the Riemannian gradient descent algorithm in Section 9.

7. Dual Connections and Alpha-Geometry on ${\mathcal{M}}_{\mathit{n}}^{\mathbf{+}}$

7.1. The Alpha-Connection on ${\mathcal{M}}_n^{+}$

The full power of information geometry emerges through Amari’s family of $\alpha$-connections [5,29], which interpolate between the exponential ($\alpha =1$) and mixture ($\alpha =-1$) connections and share the Levi-Civita connection as their self-dual midpoint at $\alpha =0$. For the Gaussian family parameterised by the precision matrix $S\in {\mathrm{SPD}}_n$, these connections are evaluable using a finite algorithm from matrix input: their Christoffel symbols are rational functions of S and $S^{-1}$ (for the Fisher–Rao connection) or polynomial in the natural parameters and the derivatives of the log-partition function (for the $\alpha$-connections), and are thus algorithmically computable—given rational-entry representations of a point and two tangent vectors, the symbols can be evaluated to arbitrary precision by a finite sequence of arithmetic operations.

Definition 8

(Alpha-Connections on ${\mathcal{M}}_n^{+}$). The α-connection on ${\mathcal{M}}_n^{+}$ is defined, on the symmetric factor, as the α-connection of the Gaussian model:

$${\mathsf{\Gamma }}_{jk}^{\left(\alpha \right),i}\left(S\right)=-\frac{1+\alpha }{2}{\left[{\left(g_S^F\right)}^{-1}·{\partial }_j{\partial }_k\log p(\mathbf{x};S)\right]}^i,$$

(33)

where $p(\mathbf{x};S)={\left(2\pi \right)}^{-n/2}{\left|S\right|}^{1/2}\exp (-{\mathbf{x}}^{T}S\mathbf{x}/2)$ is the density of $\mathcal{N}(0,S^{-1})$. On the skew factor, the α-connection is the standard flat connection (all Christoffel symbols zero).

For the Gaussian model, the natural parameters are the entries of the precision matrix $S\in {\mathrm{SPD}}_n$, and the model is an exponential family. Exponential families have the important property that the e-connection ($\alpha =1$) is flat [5]: the Christoffel symbols of ${\nabla }^{\left(1\right)}$ vanish in natural coordinates.

Theorem 9

(Flatness of the Exponential Connection on ${\mathcal{M}}_n^{+}$). The $\alpha =1$ (exponential) connection ${\nabla }^{\left(1\right)}$ on the symmetric factor of ${\mathcal{M}}_n^{+}$ is flat with respect to the natural coordinates ${\left(S_{ij}\right)}_{i\le j}$ of ${\mathrm{SPD}}_n$. Equivalently, the S-factor of ${\mathcal{M}}_n^{+}$ is a dually flat manifold with potential function $\psi \left(S\right)=-{\displaystyle \frac{1}{2}}\log \left|S\right|+{\displaystyle \frac{n}{2}}\log \left(2\pi \right)$ (the log-partition function of the Gaussian family).

Proof. 

The Gaussian model $\{p(\mathbf{x};S):S\in {\mathrm{SPD}}_n\}$ is a full exponential family in canonical form $p(\mathbf{x};\theta )=\exp (\langle \theta ,T(\mathbf{x})\rangle -\psi (\theta \left)\right)$, where $\theta =S/2$ (the natural parameter) and $T\left(\mathbf{x}\right)=-\mathbf{x}{\mathbf{x}}^T$ (the sufficient statistic, a symmetric matrix). For any exponential family, the e-connection is flat and the ${\nabla }^{\left(1\right)}$-geodesics are straight lines in natural coordinates [5]. The log-partition function is $\psi \left(\theta \right)=-{\displaystyle \frac{1}{2}}\log \left|2\theta \right|+{\displaystyle \frac{n}{2}}\log \left(2\pi \right)=-{\displaystyle \frac{1}{2}}\log \left|S\right|+{\displaystyle \frac{n}{2}}\log \left(2\pi \right)$. □

Corollary 2

(Dual Flat Structure). The symmetric factor $({\mathrm{SPD}}_n,g^F,{\nabla }^{\left(1\right)},{\nabla }^{(-1)})$ is a dually flat statistical manifold. The e-geodesics (paths straight in natural parameters S) are Euclidean straight lines $S\left(t\right)=(1-t)S_0+t{S}_1$. The m-geodesics (paths straight in mean parameters $\mathsf{\Sigma }=\mathbb{E}\left[\mathbf{x}{\mathbf{x}}^T\right]=S^{-1}$) are straight lines in the covariance matrix: $\mathsf{\Sigma }\left(t\right)=(1-t){\mathsf{\Sigma }}_0+t{\mathsf{\Sigma }}_1$.

7.2. The Bregman Divergence and Stability

On a dually flat manifold, the natural divergence is the Bregman divergence associated with the potential function. For the Gaussian precision matrix model:

$$D_{\psi }(S_0\parallel S_1)=\psi \left(S_0\right)-\psi \left(S_1\right)-{\langle \nabla \psi \left(S_1\right),S_0-S_1\rangle }_{g^F}=\frac{1}{2}{L}_S(S_0,S_1),$$

(34)

that is, the Bregman divergence of $\psi$ is one half of the Stein loss (13), and equivalently the Kullback–Leibler divergence $\mathrm{KL}\left(\mathcal{N}(0,S_1^{-1})\parallel \mathcal{N}(0,S_0^{-1})\right)$ between the corresponding Gaussians [24]. All subsequent uses of $D_{\psi }$ follow this $\frac{1}{2}$ convention.

Proposition 9

(Stein Loss as a Stability Monitor). Let $A_0\in {\mathcal{M}}_n^{+}$ with $\omega \left(A_0\right)<0$. For a perturbation $A_1=A_0+\delta A$ with $A_1\in {\mathcal{M}}_n^{+}$, the Stein loss between the symmetric parts is

$$D_{\psi }(S\left(A_0\right)\parallel S\left(A_1\right))\approx \frac{1}{2}\mathrm{tr}\left({\left(S_0^{-1}\delta S\right)}^2\right)=\frac{1}{2}{∥S_0^{-1/2}\delta S{S}_0^{-1/2}∥}_F^2$$

(35)

to the second order in $\delta S=S\left(\delta A\right)$. The condition $D_{\psi }(S_0\parallel S_1)\le \epsilon$ implies ${\lambda }_{\max }\left(S_1\right)\le {\lambda }_{\max }\left(S_0\right)+\sqrt{2\epsilon }{∥S_0∥}_2$, preserving the stability property $\omega \left(A_1\right)<0$ as long as $\epsilon <\omega {\left(A_0\right)}^2/\left(2{∥S_0∥}_2^2\right)$.

The precise relationship between the Stein loss and the squared Fisher–Rao distance, and the computational advantage of the former, are set out in Appendix C.

Proof. 

The second-order expansion of (34) around $S_1=S_0+\delta S$ gives $D_{\psi }(S_0\parallel S_0+\delta S)={\displaystyle \frac{1}{2}}\mathrm{tr}\left({\left(S_0^{-1}\delta S\right)}^2\right)+{O(\parallel \delta S\parallel }^3)$ by Taylor expansion of $\log \det$ and the trace formula. For the stability implication, $|{\lambda }_{\max }(S_0+\delta S)-{\lambda }_{\max }\left(S_0\right)|\le {∥\delta S∥}_2\le {∥\delta S∥}_F\le {∥S_0∥}_2{∥S_0^{-1/2}\delta S{S}_0^{-1/2}∥}_F\le {∥S_0∥}_2\sqrt{2\epsilon }$, so stability persists as long as $|\omega \left(A_0\right)|>\sqrt{2\epsilon }{∥S_0∥}_2$. □

7.3. The Legendre Transform and Mean-Parameter Coordinates

The duality between e-coordinates (natural parameters S) and m-coordinates (mean parameters $\mathsf{\Sigma }=S^{-1}$) is mediated by the Legendre transform of the potential $\psi \left(S\right)=-{\displaystyle \frac{1}{2}}\log \left|S\right|$:

$$\varphi \left(\mathsf{\Sigma }\right)=\underset{S\in {\mathrm{SPD}}_n}{\sup }\left[{\langle S,\mathsf{\Sigma }\rangle }_F-\psi \left(S\right)\right]=\frac{1}{2}\log \left|\mathsf{\Sigma }\right|+\frac{n}{2}(1+\log \left(2\pi \right)),$$

(36)

where $\varphi$ is the dual potential (the entropy of $\mathcal{N}(0,\mathsf{\Sigma })$). The Legendre transform encodes the passage from precision matrix coordinates (natural for stability analysis; $S\succ 0$ is the stability condition) to covariance matrix coordinates (natural for estimation; $\mathsf{\Sigma }$ is the sample covariance). This duality is particularly useful in the financial application (Section 10), where the estimated quantity is the sample covariance $\widehat{\mathsf{\Sigma }}$ but the stability condition is on $S={\widehat{\mathsf{\Sigma }}}^{-1}$ (or equivalently on the drift matrix A).

8. Pseudospectrum and Information-Geometric Certificates

8.1. Pseudospectrum as a Distance in ${\mathcal{M}}_n^{+}$

Recall the $\epsilon$-pseudospectrum ${\mathsf{\Lambda }}_{\epsilon }\left(A\right)$ of (11). Each shifted matrix $zI-A$ is itself a complex matrix; restricting to $z\in {\mathbb{R}}_{\mathrm{c}}$, we obtain the set $\{x\in {\mathbb{R}}_{\mathrm{c}}:{\sigma }_{\min }(xI-A)\le \epsilon \}$, which is a union of intervals covering the real parts of the $\epsilon$-pseudospectrum.

Theorem 10

(Pseudospectrum via Distance in ${\mathcal{M}}_n^{+}$). For $A\in {\mathcal{M}}_n^{+}$ and $x\in {\mathbb{R}}_{\mathrm{c}}$, the minimal singular value of $xI-A$ satisfies

$${\sigma }_{\min }(xI-A)={\sqrt{{\lambda }_{\min }\left(S{(xI-A)}^2+K{\left(A\right)}^2-[S(xI-A),K\left(A\right)]\right)}}^{1/2},$$

(37)

and in particular ${\sigma }_{\min }(xI-A)\le \left|x-\omega \left(A\right)\right|$ for all $x\in {\mathbb{R}}_{\mathrm{c}}$, with equality when A is normal.

Proof. 

The symmetric part of $xI-A$ is $(x-S(A\left)\right)$ and the skew part is $-K\left(A\right)$. The smallest singular value satisfies ${\sigma }_{\min }^2={\lambda }_{\min }\left({(xI-A)}^T(xI-A)\right)={\lambda }_{\min }({(xI-S\left(A\right))}^2+K{\left(A\right)}^2+[(xI-S\left(A\right)),K\left(A\right)])$. The commutator $\left[\right(xI-S),K]=-[S,K]$ (since $[xI,K]=0$). For the bound, using ${\sigma }_{\min }^2\le {\lambda }_{\min }\left(S{(xI-A)}^{T}S(xI-A)\right)={(x-{\lambda }_{\max }\left(S\left(A\right)\right))}^2$ when $x>{\lambda }_{\max }\left(S\right)$, which gives ${\sigma }_{\min }(xI-A)\le x-\omega \left(A\right)$. Equality for normal A follows from Theorem 1. □

Corollary 3

(Kreiss Constant Bound). The Kreiss constant of $A\in {\mathcal{M}}_n^{+}$ satisfies

$$\mathcal{K}\left(A\right)≔\underset{x>0}{\sup }x{∥{(xI+A)}^{-1}∥}_2\le e^{|m^F\left(A\right)|},$$

(38)

where $|m^F\left(A\right)|=|\log {\lambda }_{\min }(-S\left(A\right))|=|\log \left|\omega \right(A\left)\right||$ is the magnitude of the Fisher–Rao stability margin.

Proof. 

When $\omega \left(A\right)<0$, the resolvent ${(xI+A)}^{-1}$ satisfies ${∥{(xI+A)}^{-1}∥}_2{\le 1/(x+\omega \left(A\right)-\mathcal{C}\left(A\right)}^2/\left(8\right|\omega \left(A\right)|{∥S∥}_F^2\left)\right)$ by the bound of Theorem 7. Taking the supremum over $x>0$ and exponentiating the margin magnitude gives (38). □

8.2. Pseudospectral Width and Non-Normality

The pseudospectral abscissa ${\alpha }_{\epsilon }\left(A\right)≔\sup \{\mathrm{Re}\left(z\right):z\in {\mathsf{\Lambda }}_{\epsilon }\left(A\right)\}$ measures how far the pseudospectrum extends into the right half-plane. For a stable A ($\alpha <0$), a large $\epsilon$-pseudospectral abscissa indicates that small perturbations can render the system unstable.

Proposition 10

(Pseudospectral Width and Commutator). For $A\in {\mathcal{M}}_n^{+}$ with $\omega \left(A\right)<0$, the pseudospectral abscissa satisfies

$${\alpha }_{\epsilon }\left(A\right)\le \omega \left(A\right)+\epsilon +{\displaystyle \frac{\mathcal{C}\left(A\right)}{2{∥S\left(A\right)∥}_F}},$$

(39)

where $\mathcal{C}\left(A\right)={∥\left[S\right(A),K(A\left)\right]∥}_F$. In particular, for a normal A (where $\mathcal{C}=0$), ${\alpha }_{\epsilon }\left(A\right)\le \alpha \left(A\right)+\epsilon$.

Proof. 

By definition, ${\alpha }_{\epsilon }\left(A\right)={\max }_{∥E∥\le \epsilon }\alpha (A+E)$. For any E with ${∥E∥}_2\le \epsilon$, $\omega (A+E)\le \omega \left(A\right)+{∥S\left(E\right)∥}_2\le \omega \left(A\right)+\epsilon$. The correction term $\mathcal{C}\left(A\right)/\left(2{∥S∥}_F\right)$ arises from the non-normality gap $\omega (A+E)-\alpha (A+E)\le \mathcal{C}(A+E)/\left(2{∥S(A+E)∥}_F\right)\approx \mathcal{C}\left(A\right)/\left(2{∥S\left(A\right)∥}_F\right)$ to leading order in E. □

9. Computational Algorithms

9.1. Algorithm 1: Fisher–Rao Stability Assessment

The following algorithm computes the stability certificate for a given interaction matrix $A\in {\mathbb{R}}_{\mathrm{c}}^{n\times n}$, returning the numerical abscissa $\omega \left(A\right)$, the Fisher–Rao stability margin $m^F\left(A\right)$, and the non-normality defect $\mathcal{C}\left(A\right)$.

• Time complexity. We first establish the running time of Algorithm 1 rigorously, then relate it to time-bounded Kolmogorov complexity.

Lemma 2

(Running Time of Algorithm 1). On an input matrix $A\in {\mathbb{R}}_{\mathrm{c}}^{n\times n}$, Algorithm 1 terminates after $\mathsf{\Theta }\left(n^3\right)$ arithmetic operations and uses $O\left(n^2\right)$ working memory.

Proof. 

We account for each step. Forming $S=\frac{1}{2}(A+A^{\mathsf{T}})$ and $K=\frac{1}{2}(A-A^{\mathsf{T}})$ costs $\mathsf{\Theta }\left(n^2\right)$ additions. The symmetric eigenvalue decomposition of S (Step 2) by the standard Householder tridiagonalisation followed by the symmetric QR or divide-and-conquer iteration costs $\mathsf{\Theta }\left(n^3\right)$ arithmetic operations [9] (§8.3). Extracting $\omega \left(A\right)={\lambda }_{\max }\left(S\right)$ and ${\lambda }_{\min }(-S)$ from the computed spectrum is $O\left(n\right)$. The commutator $C=SK-KS$ requires two $n\times n$ matrix products (Step 10), each $\mathsf{\Theta }\left(n^3\right)$ by the schoolbook algorithm, and one $\mathsf{\Theta }\left(n^2\right)$ subtraction; its Frobenius norm is $O\left(n^2\right)$. All remaining steps (scalar logarithms, comparisons, norm evaluations) are $O\left(n^2\right)$. In sum, the dominant contributions are the $\mathsf{\Theta }\left(n^3\right)$ eigendecomposition and the $\mathsf{\Theta }\left(n^3\right)$ products, giving $\mathsf{\Theta }\left(n^3\right)$ total operations; no step allocates more than a constant number of $n\times n$ arrays, so the memory is $O\left(n^2\right)$. □

Algorithm 1 Information-Geometric Stability Assessment

Require: $A\in {\mathbb{R}}_{\mathrm{c}}^{n\times n}$ Ensure: $\omega$, $m_F$, $\mathcal{C}$, stability flag   1: $S\leftarrow (A+A^T)/2$; $K\leftarrow (A-A^T)/2$   2: Compute eigenvalues ${\lambda }_1\ge \cdots \ge {\lambda }_n$ of S via symmetric eigensolver (cost: $O\left(n^3\right)$, backward error $O\left({\epsilon }_{\mathrm{mach}}\right){∥S∥}_2$)   3: $\omega \leftarrow {\lambda }_1$                            ▹${\lambda }_{\max }\left(S\right)$   4: if $\omega <0$ then   5:     stable ← true   6:     $m_F\leftarrow \log \left|\omega \right|$        ▹ stability margin $\log {\lambda }_{\min }(-S)$, Equation (20)   7: else   8:     stable ←false;  $d_F\leftarrow 0$   9: end if 10: Compute $C\leftarrow SK-KS$                 ▹$n^2$ multiplications 11: $\mathcal{C}\leftarrow {∥C∥}_F$ 12: Compute transient bound $\mathcal{T}\leftarrow \exp ({\mathcal{C}}^2/\left(8\right|\omega |{∥S∥}_F^2\left)\right)$ if stable, else $\mathcal{T}\leftarrow +\infty$ 13: return $\omega$, $d_F$, $\mathcal{C}$, stable, $\mathcal{T}$

• Relation to Kolmogorov complexity. We can now state precisely how this running time relates to Kolmogorov complexity, correcting the imprecise claim of “unrelatedness” in the previous version. The plain Kolmogorov complexity $K\left(x\right)$ of a finite string x is the length of the shortest program (on a fixed universal machine U) that outputs x; it is descriptional, universal up to an additive constant by the invariance theorem, and uncomputable [30]. The bridge to running time is the time-bounded Kolmogorov complexity

  $$K^t\left(x\right)=\min \left\{\right|q|:U\left(q\right)=x\mathrm{in}\mathrm{at}\mathrm{most}t\mathrm{steps}\},$$

  (40)

  which augments the descriptional measure with an explicit time budget t and satisfies $K^t\left(x\right)\ge K\left(x\right)$ for every t, with $K^t\left(x\right)\downarrow K\left(x\right)$ as $t\to \infty$; this monotone relation is exactly the link between the two notions, since enlarging the time resource t relaxes $K^t$ towards the unbounded descriptional optimum K.

We make the connection to Algorithm 1 quantitative. Fix the dimension n and let A be a rational-entry input encoded by a string of length $\ell \left(A\right)$. Let $q_{\star }$ be the fixed program implementing Algorithm 1, of constant length $|q_{\star }|=c_0$ independent of n. Running $q_{\star }$ on the encoded pair $(A,2^{-m})$ produces the output $d^F\left(A\right)$ to precision $2^{-m}$, and by Lemma 2, it does so within

$$t(n,m)=\mathsf{\Theta }\left(n^3\right)·\mathrm{poly}(m,\ell \left(A\right))$$

(41)

steps, the second factor accounting for the bit-cost of the arithmetic on m-bit-precision rationals. Consequently, the time-bounded complexity of the output is bounded by the description length of program plus input,

$$K^{t(n,m)}\left(d^F\left(A\right){\upharpoonright }_m\right)\le c_0+\ell \left(A\right)+2\log m+O\left(1\right),$$

(42)

where $d^F\left(A\right){\upharpoonright }_m$ denotes the output truncated to m bits and the $2\log m$ term encodes the precision parameter. Thus, the $O\left(n^3\right)$ time bound of Lemma 2 is exactly the resource t appearing in the exponent of $K^t$ in (42): the algorithm’s time complexity is the time budget that makes the output’s time-bounded Kolmogorov complexity as small as the trivial program-plus-input description. Plain (unbounded) Kolmogorov complexity $K\left(d^F\left(A\right)\right)$ remains a distinct, uncomputable quantity bounded above by the right-hand side of (42); it is the time-bounded version $K^t$, through the budget $t=\mathsf{\Theta }\left(n^3\right)$, that is the correct point of contact between the descriptive and the computational-time notions of complexity.

• Numerical stability. The symmetric eigensolver in Step 2 has backward error $O\left({\epsilon }_{\mathrm{mach}}{∥S∥}_2\right)$ with no condition-number amplification (since S is symmetric). The commutator $C=SK-KS$ is computed via two standard matrix products; its Frobenius norm is bounded by ${∥C∥}_F\le 2{∥S∥}_2{∥K∥}_F$, so no cancellation issues arise.

9.2. Algorithm 2: Riemannian Gradient Descent for Stable Matrix Estimation

The information-geometric regularisation problem (44) is a constrained optimisation on the Riemannian manifold $({\mathrm{SPD}}_n,g^F)$. Riemannian gradient descent (RGD) exploits the manifold structure via the exponential map (31) and the Riemannian gradient.

Algorithm 2 Riemannian Gradient Descent for Stable Drift Estimation

Require: Estimated drift $\widehat{A}\in {\mathbb{R}}_{\mathrm{c}}^{n\times n}$, regularisation $\lambda >0$, step size $\eta >0$, tolerance $\tau >0$ Ensure: Regularised stable drift $A^{\mathrm{reg}}$   1: Initialise $S^{\left(0\right)}\leftarrow S\left(\widehat{A}\right)$; $K^{\left(0\right)}\leftarrow K\left(\widehat{A}\right)$   2: while not converged do   3:     Compute Euclidean gradient of $f\left(S\right)=d^F{(S,S\left(\widehat{A}\right))}^2+\lambda {\lambda }_{\max }{\left(S\right)}_{+}$ at $S^{\left(t\right)}$: $$\begin{array}{cc}\hfill {\nabla }_{S}f& =2{\log }_{S^{\left(t\right)}}\left(S\left(\widehat{A}\right)\right)+\lambda {\mathbf{u}}_1{\mathbf{u}}_1^T·\mathbf{1}[\omega >0],\hfill \end{array}$$ where ${\mathbf{u}}_1$ is the leading eigenvector of $S^{\left(t\right)}$   4:     Convert to Riemannian gradient: ${\mathrm{grad}}_{g}f=S^{\left(t\right)}{\nabla }_{S}f{S}^{\left(t\right)}$   ▹ via $g^F$ metric tensor   5:     Update: $S^{(t+1)}\leftarrow {\exp }_{S^{\left(t\right)}}\left(-\eta {\mathrm{grad}}_{g}f\right)$ via Equation (31)   6:     $K^{(t+1)}\leftarrow K^{\left(t\right)}$                    ▹ skew part unchanged   7:     if ${∥S^{(t+1)}-S^{\left(t\right)}∥}_F<\tau$ then break   8:     end if   9: end while 10: $A^{\mathrm{reg}}\leftarrow S^{(t+1)}+K^{(t+1)}$ 11: return $A^{\mathrm{reg}}$

• Convergence. Since f is the sum of a geodesically convex function ($d^F{(·,\widehat{S})}^2$ is geodesically convex on the Hadamard manifold $({\mathrm{SPD}}_n,g^F)$; see [16]) and a convex function (${\lambda }_{\max }(·)$, which is geodesically convex on ${\mathrm{SPD}}_n$ [31]), the objective f is geodesically convex, and Riemannian gradient descent converges to the global minimum at a rate of $O(1/t)$ [32].

Proposition 11

(Convergence Rate of Algorithm 2). Let $f:{\mathrm{SPD}}_n\to {\mathbb{R}}_{\mathrm{c}}$ be L-smooth and geodesically convex. With step size $\eta =1/L$, Algorithm 2 satisfies $f\left(S^{\left(T\right)}\right)-f\left(S^{*}\right)\le O(R^{2}L/T)$, where $R=d^F(S^{\left(0\right)},S^{*})$ is the initial distance to the optimum $S^{*}$ and L is the Lipschitz constant of $\nabla f$.

Proof. 

The result follows from the standard Riemannian convex convergence analysis on Hadamard manifolds; see [32] Theorem 4.1 for the precise statement. The smoothness constant L for $d^F{(·,\widehat{S})}^2$ is $L=2/{\lambda }_{\min }{\left(S\left(\widehat{A}\right)\right)}^2$, and for ${\lambda }_{\max }(·)$, it is bounded by the inverse spectral gap. □

9.3. Algorithm 3: Commutator-Regularised Neural Network Training

Theorem 12 suggests that penalising the commutator norm $\mathcal{C}\left(\mathbf{J}\right)$ during training reduces transient gradient risk. The following loss augmentation implements this.

Algorithm 3 Commutator-Regularised Backpropagation

Require: Mini-batch $\mathcal{B}$, model parameters $\theta$, regularisation $\mu >0$, layer index ℓ Ensure: Updated parameters ${\theta }^{\prime }$ 1: Compute Jacobian ${\mathbf{J}}_{\ell }=\mathrm{diag}\left({\sigma }_{\ell }^{\prime }\right){\mathbf{W}}_{\ell }$ for layer ℓ     ▹$O\left(nd\right)$ cost 2: $S_{\ell }\leftarrow ({\mathbf{J}}_{\ell }+{\mathbf{J}}_{\ell }^T)/2$; $K_{\ell }\leftarrow ({\mathbf{J}}_{\ell }-{\mathbf{J}}_{\ell }^T)/2$ 3: ${\mathcal{C}}_{\ell }\leftarrow {∥S_{\ell }{K}_{\ell }-K_{\ell }{S}_{\ell }∥}_F$ 4: Augmented loss: ${\mathcal{L}}_{\mathrm{aug}}={\mathcal{L}}_{\mathrm{task}}+\mu {\mathcal{C}}_{\ell }^2$ 5: Compute gradient ${\nabla }_{\theta }{\mathcal{L}}_{\mathrm{aug}}$ via backpropagation, including the commutator gradient: $${\nabla }_{{\mathbf{W}}_{\ell }}{\mathcal{C}}_{\ell }^2=4\mathrm{diag}\left({\sigma }_{\ell }^{\prime }\right)\left([S_{\ell },K_{\ell }]\right)+4\left[[S_{\ell },K_{\ell }],\mathrm{diag}\left({\sigma }_{\ell }^{\prime }\right)\right]{\mathbf{W}}_{\ell }$$ 6: Update ${\theta }^{\prime }\leftarrow \theta -\eta {\nabla }_{\theta }{\mathcal{L}}_{\mathrm{aug}}$ 7: return ${\theta }^{\prime }$

The commutator regularisation in Algorithm 3 adds $O\left(n{d}^2\right)$ overhead per layer per mini-batch, which is $O\left(d\right)$ times the cost of computing the Jacobian itself. For practical deep networks with $d\le 1024$, this is computationally feasible in the backward pass.

10. Applications

10.1. Ecological Community Matrices and May’s Criterion

May [10] considered random community matrices of the form $A=-dI+\delta C$, where $d>0$ is a self-damping constant, $\delta >0$ is the interaction strength, and $C=\left(c_{ij}\right)$ is an $n\times n$ matrix with i.i.d. entries $c_{ij}\sim \mathcal{N}(0,1)$ (with $c_{ii}=0$). May’s celebrated result is that such a community is stable with high probability as $n\to \infty$ if and only if $\delta \sqrt{n}<d$.

In information-geometric terms, $S\left(A\right)=-dI+\delta S\left(C\right)$ and $K\left(A\right)=\delta K\left(C\right)$. The symmetric part $S\left(C\right)=(C+C^{\mathsf{T}})/2$ has entries $(c_{ij}+c_{ji})/2\sim \mathcal{N}(0,1/2)$ for $i\ne j$, so by the Wigner semicircle law [33], ${\lambda }_{\max }\left(S\left(C\right)\right)\approx \sqrt{2n}·(1/\sqrt{2})=\sqrt{n}$ with high probability as $n\to \infty$. Therefore,

$$\omega \left(A\right)=-d+\delta {\lambda }_{\max }\left(S\left(C\right)\right)\approx -d+\delta \sqrt{n}.$$

(43)

May’s criterion $\delta \sqrt{n}<d$ is precisely the condition $\omega \left(A\right)<0$, i.e., $A\in {\mathcal{M}}_n^{+}$ with $S\left(A\right)\prec 0$. This establishes the following information-geometric reformulation.

Theorem 11

(May’s Criterion via Information Geometry). Let $A=-dI+\delta C$ be a random community matrix as above. The Fisher–Rao stability margin satisfies $m^F\left(A\right)=\log {\lambda }_{\min }(-S\left(A\right))=\log (d-\delta \sqrt{n})+o\left(1\right)$ asymptotically as $n\to \infty$. The community is stable with high probability if and only if this margin is finite with ${\lambda }_{\min }(-S\left(A\right))>0$, i.e., if and only if the precision matrix $-S\left(A\right)$ lies in the interior of ${\mathrm{SPD}}_n$; equivalently, $\delta \sqrt{n}<d$.

Proof. 

The eigenvalues of $S\left(A\right)=-dI+\delta S\left(C\right)$ are $-d+\delta {\mu }_i$, where ${\mu }_i$ are the eigenvalues of $S\left(C\right)$. By the Wigner semicircle law, ${\mu }_i\in [-\sqrt{2n},\sqrt{2n}]$ asymptotically. The stability condition ${\lambda }_{\max }\left(S\left(A\right)\right)<0$ is $-d+\delta \sqrt{2n}/\sqrt{2}<0$, i.e., $\delta \sqrt{n}<d$ (adjusting the constant appropriately for the specific entry distribution). Hence, ${\lambda }_{\min }(-S\left(A\right))=d-\delta \sqrt{n}+o\left(1\right)$, and the stability margin is $m^F\left(A\right)=\log {\lambda }_{\min }(-S\left(A\right))=\log (d-\delta \sqrt{n})+o\left(1\right)$, which is positive precisely when $\delta \sqrt{n}<d$. □

This result extends to structured community matrices. Allesina and Tang [11] showed that for matrices with a specific predator–prey sign structure, stability conditions differ from May’s criterion. The information-geometric framework captures this: the sign structure of C affects the distribution of the eigenvalues of $S\left(C\right)$, and hence the numerical abscissa $\omega \left(A\right)$. Allesina et al. [28] further demonstrated that the structure of interactions (predator–prey vs. mutualism vs. competition) systematically shifts the spectral radius under the circular law [22,23], and these shifts are reflected in the Fisher–Rao distance to the instability boundary.

The skew-symmetric part $K\left(A\right)=\delta K\left(C\right)$ encodes the net directed flow in the food web. In the Lotka–Volterra framework [34,35], the community matrix for a predator–prey pair $(i,j)$ has $a_{ij}={\delta }_{ij}$ (prey j benefits predator i) and $a_{ji}=-{\delta }_{ij}$ (predator i harms prey j), so $K\left(A\right)$ is exactly the anti-symmetric interaction matrix. The commutator norm $\mathcal{C}\left(A\right)={∥\left[S\right(A),K(A\left)\right]∥}_F$ measures the coupling between the self-damping dynamics and the directed energy flow in the food web; by Theorem 7, a large $\mathcal{C}\left(A\right)$ predicts large transient population fluctuations even in a globally stable ecosystem.

10.2. Financial Covariance Matrices and Correlation Networks

In financial applications, the interaction matrix A arises as the instantaneous drift of a multivariate diffusion model $d{\mathbf{X}}_t=A{\mathbf{X}}_{t}dt+{\mathsf{\Sigma }}^{1/2}d{\mathbf{W}}_t$, where $A\in {\mathbb{R}}_{\mathrm{c}}^{n\times n}$ encodes cross-asset momentum and mean-reversion effects [27]. Empirical estimation of A from finite time series yields $\widehat{A}$, and the question of whether $\widehat{A}$ is in ${\mathcal{M}}_n^{+}$ (stable system) is critical for portfolio management.

The empirical covariance matrix $\widehat{\mathsf{\Sigma }}=T^{-1}\mathbf{X}{\mathbf{X}}^{\mathsf{T}}$ of asset returns is known to be ill-conditioned for an $n/T$ close to 1 [25]. Ledoit and Wolf [25,26] developed shrinkage estimators that improve the condition number of $\widehat{\mathsf{\Sigma }}$ by combining it with a structured target. The information-geometric perspective provides a principled target: the identity in $({\mathrm{SPD}}_n,g^F)$ is the centre of symmetry of the Fisher–Rao metric, and shrinkage toward the identity corresponds to moving along a Fisher–Rao geodesic.

The estimated drift matrix $\widehat{A}$ from historical returns has a symmetric part $S\left(\widehat{A}\right)$ that determines stability: the system is mean-reverting (stable) iff $\omega \left(\widehat{A}\right)={\lambda }_{\max }\left(S\left(\widehat{A}\right)\right)<0$. We propose the following information-geometric regularisation:

$$A^{\mathrm{reg}}={\arg \min }_{A^{\prime }\in {\mathcal{M}}_n^{+}}\left[d^F{(S\left(A^{\prime }\right),S\left(\widehat{A}\right))}^2+\lambda \omega {\left(A^{\prime }\right)}_{+}\right],$$

(44)

where $\omega {\left(A^{\prime }\right)}_{+}=\max (0,\omega \left(A^{\prime }\right))$ penalises instability and $\lambda >0$ is a regularisation parameter. The optimisation (44) is convex in $S\left(A^{\prime }\right)$ (since $d^F{(·,\widehat{S})}^2$ is convex on $({\mathrm{SPD}}_n,g^F)$ [16] and ${\lambda }_{\max }(·)$ is convex) and can be solved by projected gradient descent on ${\mathrm{SPD}}_n$.

The asymmetric part $K\left(\widehat{A}\right)$ encodes lead–lag relationships between assets: $a_{ij}\ne a_{ji}$ means asset j leads asset i (or vice versa) in the return dynamics. The commutator norm $\mathcal{C}\left(\widehat{A}\right)={∥[S\left(\widehat{A}\right),K\left(\widehat{A}\right)]∥}_F$ quantifies how strongly the lead–lag structure couples to the mean-reversion structure. By Theorem 7, a large $\mathcal{C}\left(\widehat{A}\right)$ predicts large short-run volatility amplification even if the long-run system is stable.

The Barabási–Albert random network model [36] generates scale-free interaction graphs whose adjacency matrices are highly asymmetric and non-normal. For financial correlation networks built on such topologies, the commutator norm $\mathcal{C}$ scales as $O\left(n^{3/2}\right)$ with network size, implying rapid growth of transient amplification with network connectivity—a finding consistent with empirical observations of volatility clustering in large financial networks. A correlation network (or Gaussian graphical model) is a graph whose nodes are random variables $X_1,\dots ,X_n$ and whose edges encode partial correlations. The partial correlation between $X_i$ and $X_j$ is their correlation after linearly removing the effect of all remaining variables; it vanishes if and only if $X_i$ and $X_j$ are conditionally independent given the others, meaning that the conditional distribution of $(X_i,X_j)$ given $\{X_k:k\ne i,j\}$ is a is a product of the two conditional marginals, i.e., it factorises. For a multivariate Gaussian with covariance $\mathsf{\Sigma }$, these relationships are read off the precision matrix $\mathsf{\Omega }={\mathsf{\Sigma }}^{-1}$: the partial correlation between $X_i$ and $X_j$ equals $-{\mathsf{\Omega }}_{ij}/\sqrt{{\mathsf{\Omega }}_{ii}{\mathsf{\Omega }}_{jj}}$, so a zero off-diagonal entry ${\mathsf{\Omega }}_{ij}=0$ is exactly conditional independence, and the support of $\mathsf{\Omega }$ is the edge set of the graph [37]. In finance, the nodes are assets and the edges encode partial correlations of returns after controlling for the remaining assets.

10.3. Neural Network Jacobians

For a deep feedforward network with layer function $f_{\ell }\left(\mathbf{x}\right)={\sigma }_{\ell }({\mathbf{W}}_{\ell }\mathbf{x}+{\mathbf{b}}_{\ell })$, the Jacobian of the end-to-end map at a point $\mathbf{x}$ is $\mathbf{J}={\prod }_{\ell =L}^1\mathrm{diag}\left({\sigma }_{\ell }^{\prime }\right){\mathbf{W}}_{\ell }$, which is the product of asymmetric matrices [38]. The stability of gradient backpropagation is governed by the singular values of $\mathbf{J}$, while the dynamics of recurrent networks are governed by the eigenvalues of the recurrent weight matrix $\mathbf{W}$ [39].

The information-geometric framework applies to the Jacobian $\mathbf{J}$ viewed as an element of ${\mathcal{M}}_n^{+}$ (when $S\left(\mathbf{J}\right)\succ 0$). The gradient signal $\delta ={\mathbf{J}}^{\mathsf{T}}ϵ$ (backpropagated error) has the expected squared norm

$$\mathbb{E}{∥\delta ∥}^2=\mathbb{E}\left[{ϵ}^T\mathbf{J}{\mathbf{J}}^{\mathsf{T}}ϵ\right]\le {\lambda }_{\max }\left(\mathbf{J}{\mathbf{J}}^{\mathsf{T}}\right)\mathbb{E}{∥ϵ∥}^2={\sigma }_1{\left(\mathbf{J}\right)}^2\mathbb{E}{∥ϵ∥}^2.$$

(45)

Gradient explosion (${\sigma }_1\gg 1$) and gradient vanishing (${\sigma }_n\ll 1$) are thus controlled by the singular values. The numerical abscissa provides a complementary bound on the forward dynamics: $\omega \left(\mathbf{J}\right)={\lambda }_{\max }\left(S\left(\mathbf{J}\right)\right)$ controls the maximum amplification of the forward pass signal.

Theorem 12

(Jacobian Stability Criterion). Let $\mathbf{J}=S+K\in {\mathcal{M}}_n^{+}$ be the Jacobian of a neural network layer.

(i) 

The condition $\omega \left(\mathbf{J}\right)<0$ (i.e., $S\left(\mathbf{J}\right)\prec 0$) implies that the forward dynamics are contractive, and hence the network is depth-stable: $∥\mathbf{J}\mathbf{x}∥<∥\mathbf{x}∥$ for all $\mathbf{x}\ne 0$.

(ii) 

The maximum gradient amplification satisfies ${∥{\mathbf{J}}^{\mathsf{T}}ϵ∥}_2/{∥ϵ∥}_2\le e^{\omega \left(\mathbf{J}\right)}$ when $S\left(\mathbf{J}\right)\prec 0$.

(iii) 

The non-normality defect $\mathcal{C}\left(\mathbf{J}\right)={∥\left[S\right(\mathbf{J}),K(\mathbf{J}\left)\right]∥}_F$ controls transient gradient amplification via Theorem 7, providing a curvature-based regularisation target for training: minimising $\mathcal{C}\left(\mathbf{J}\right)$ over weight matrices ${\mathbf{W}}_{\ell }$ promotes normal Jacobians and stabilises gradient flow.

Proof. 

Part (i) is immediate from Theorem 6(i) applied to $\mathbf{J}$. Part (ii) follows because for $\mathbf{x}$ and $\mathbf{J}$, ${∥\mathbf{J}\mathbf{x}∥}_2\le e^{\omega \left(\mathbf{J}\right)}{∥\mathbf{x}∥}_2$ from the exponential bound (26) evaluated at $t=1$. Part (iii) is Theorem 7 applied to $\mathbf{J}$. □

The result of Lecun, Bengio, and Hinton [40] identified the control of gradient flow as a central challenge in deep learning; the Jacobian stability criterion of Theorem 12 places this control on an information-geometric foundation. Pascanu, Mikolov, and Bengio [39] established the connection between gradient explosion and the spectral radius of the recurrent weight matrix; our framework extends this to the full non-normal case, providing the gap $\omega -\alpha$ as a measure of transient gradient risk.

10.4. Advection-Diffusion Operators and the Péclet Geometry

The advection-diffusion equation is a canonical transport model that arises in geophysical fluid dynamics, atmospheric dispersion, contaminant transport, and numerical analysis [41]. Its semidiscrete form produces an asymmetric interaction matrix whose symmetric and skew-symmetric parts have a direct physical interpretation within the framework of this paper, yielding new geometric characterisations of the Péclet number, the CFL stability condition, and the mechanism by which upwind schemes regularise near-unstable operators.

10.4.1. Setup: Semidiscrete Advection-Diffusion

Consider the one-dimensional advection-diffusion equation on $[0,L]$ with homogeneous Dirichlet boundary conditions:

$${\displaystyle \frac{\partial u}{\partial t}}=D{\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}-v{\displaystyle \frac{\partial u}{\partial x}},D>0,v\in {\mathbb{R}}_{\mathrm{c}},x\in (0,L),$$

(46)

where D is the diffusion coefficient and v is the (constant) advection velocity. Discretising on a uniform grid with n interior nodes and spacing $h=L/(n+1)$ using central differences for both terms gives the semidiscrete system $\dot{\mathbf{u}}=A_c\mathbf{u}$ with

$$A_c=-{\displaystyle \frac{D}{h^2}}\mathbf{T}-{\displaystyle \frac{v}{2h}}\mathbf{C},$$

(47)

where $\mathbf{T}=\mathrm{tridiag}(-1,2,-1)\in {\mathbb{R}}_{\mathrm{c}}^{n\times n}$ is the discrete Laplacian and $\mathbf{C}=\mathrm{tridiag}(-1,0,1)\in {\mathbb{R}}_{\mathrm{c}}^{n\times n}$ is the skew-symmetric central-difference advection operator. The symmetric and skew-symmetric parts decompose as

$$S\left(A_c\right)=-{\displaystyle \frac{D}{h^2}}\mathbf{T},K\left(A_c\right)=-{\displaystyle \frac{v}{2h}}\mathbf{C}.$$

(48)

Since $\mathbf{T}$ is symmetric positive-definite (eigenvalues $2-2\cos (k\pi /(n+1\left)\right)>0$ for $k=1,\dots ,n$), the symmetric part $S\left(A_c\right)=-(D/h^2)\mathbf{T}$ is symmetric negative-definite, so $A_c\in {\mathcal{M}}_n^{+}$ for all $D>0$ and all $v\in {\mathbb{R}}_{\mathrm{c}}$. Since $\mathbf{C}$ is exactly skew-symmetric (${\mathbf{C}}^T=-\mathbf{C}$), the decomposition (48) is exact and not merely approximate.

10.4.2. The Mesh Péclet Number as a Non-Normality Index

The mesh Péclet number ${\mathrm{Pe}}_h=\left|v\right|h/\left(2D\right)$ is the classical dimensionless parameter governing the relative strength of advection and diffusion at the grid scale. We show that it is precisely a ratio of Frobenius norms on the manifold ${\mathcal{M}}_n^{+}$.

Theorem 13

(Péclet Number as Geometric Non-Normality Ratio). For the central-difference advection-diffusion matrix (47), the mesh Péclet number satisfies

$${\mathrm{Pe}}_h={\displaystyle \frac{\left|v\right|h}{2D}}={\displaystyle \frac{\parallel K\left(A_c\right){\parallel }_F}{\parallel S\left(A_c\right){\parallel }_F}}·{\displaystyle \frac{{\parallel \mathbf{T}\parallel }_F}{{\parallel \mathbf{C}\parallel }_F}}.$$

(49)

In particular, ${\mathrm{Pe}}_h={\gamma }_n·\parallel K\left(A_c\right){\parallel }_F/{\parallel S\left(A_c\right)\parallel }_F$, where the geometric factor ${\gamma }_n={\parallel \mathbf{T}\parallel }_F/{\parallel \mathbf{C}\parallel }_F$ satisfies ${\gamma }_n\to \sqrt{3}$ as $n\to \infty$. Consequently:

$$\underset{n\to \infty }{\lim }{\mathrm{Pe}}_h=\sqrt{3}{\displaystyle \frac{\parallel K\left(A_c\right){\parallel }_F}{\parallel S\left(A_c\right){\parallel }_F}}.$$

(50)

Proof. 

Direct calculation: ${\parallel \mathbf{T}\parallel }_F^2=n·4+2(n-1)·1=6n-2$, so ${\parallel \mathbf{T}\parallel }_F=\sqrt{6n-2}$. For $\mathbf{C}=\mathrm{tridiag}(-1,0,1)$, ${\parallel \mathbf{C}\parallel }_F^2=2(n-1)·1$, so ${\parallel \mathbf{C}\parallel }_F=\sqrt{2(n-1)}$. Therefore, $\parallel S\left(A_c\right){\parallel }_F=(D/h^2)\sqrt{6n-2}$ and $\parallel K\left(A_c\right){\parallel }_F=\left(\right|v|/2h)\sqrt{2(n-1)}$, giving

$${\displaystyle \frac{\parallel K\left(A_c\right){\parallel }_F}{\parallel S\left(A_c\right){\parallel }_F}}={\displaystyle \frac{\left|v\right|h}{2D}}·{\displaystyle \frac{\sqrt{2(n-1)}}{\sqrt{6n-2}}}={\mathrm{Pe}}_h·{\displaystyle \frac{{\parallel \mathbf{C}\parallel }_F}{{\parallel \mathbf{T}\parallel }_F}}.$$

Rearranging gives (49). Since ${\parallel \mathbf{T}\parallel }_F/{\parallel \mathbf{C}\parallel }_F=\sqrt{(6n-2)/(2n-2)}\to \sqrt{3}$ as $n\to \infty$, we obtain (50). □

Theorem 13 establishes that the mesh Péclet number is not merely an analogy for non-normality—it is the non-normality ratio ${\parallel K\parallel }_F/{\parallel S\parallel }_F$ up to a universal geometric constant. The non-normality defect of the operator (Definition 7) satisfies

$$\mathcal{C}\left(A_c\right)=\parallel [S\left(A_c\right),K\left(A_c\right)]{\parallel }_F={\displaystyle \frac{D\left|v\right|}{h^3}}{\parallel [\mathbf{T},\mathbf{C}]\parallel }_F,$$

(51)

and since $[\mathbf{T},\mathbf{C}]$ is a non-zero pentadiagonal matrix with ${\parallel [\mathbf{T},\mathbf{C}]\parallel }_F=O\left(n^{1/2}\right)$, the commutator norm scales as

$$\mathcal{C}\left(A_c\right)=O\left({\displaystyle \frac{D\left|v\right|}{h^3}}·n^{1/2}\right)=O\left({\displaystyle \frac{D\left|v\right|}{h^3{L}^{1/2}}}\right)\mathrm{as}n\to \infty .$$

(52)

This grows as $h^{-3}$ as the mesh is refined, showing that grid refinement increases the non-normality-induced transient amplification bound even as the truncation error decreases—a fundamental tension in numerical analysis of non-normal advection-dominated transport, here captured geometrically.

10.4.3. Fisher–Rao Stability Margin and the CFL Condition

The stability boundary $\partial {\mathcal{M}}_n^{+}=\{A:\omega \left(A\right)=0\}$ in the information-geometric sense corresponds to the onset of instability. For the advection-diffusion operator, $\omega \left(A_c\right)={\lambda }_{\max }\left(S\left(A_c\right)\right)<0$ for all $D>0$, so $A_c$ is unconditionally stable in the continuous-time sense, regardless of the Péclet number.

The Fisher–Rao stability margin of Proposition 5 evaluates to

$$m^F\left(A_c\right)=\log {\lambda }_{\min }(-S\left(A_c\right))=\left|\log \left({\displaystyle \frac{D}{h^2}}{\lambda }_{\min }\left(\mathbf{T}\right)\right)\right|=\left|\log \left({\displaystyle \frac{4D}{h^2}}{\sin }^2{\displaystyle \frac{\pi }{2(n+1)}}\right)\right|,$$

(53)

where ${\lambda }_{\min }\left(\mathbf{T}\right)=4{\sin }^2(\pi /\left(2(n+1)\right))\approx {\pi }^2{h}^2/L^2$ for fine meshes. Substituting gives

$$m^F\left(A_c\right)\approx \log \left({\displaystyle \frac{D{\pi }^2}{L^2}}\right),$$

(54)

which in the continuum limit $h\to 0$ depends only on D and L—it is a property of the continuous PDE, not the discretisation. The stability margin $d^F$ is therefore preserved under mesh refinement for the diffusion-dominated case, confirming that the Fisher–Rao distance captures continuous stability rather than purely discrete artefacts.

Now, consider the fully discrete explicit Euler scheme ${\mathbf{u}}^{m+1}=(I+\Delta t{A}_c){\mathbf{u}}^m$. The amplification matrix is $\mathbf{G}=I+\Delta t{A}_c$, and the scheme is stable in the ${\ell }^2$-norm if and only if $\rho \left(\mathbf{G}\right)\le 1$, i.e., ${\lambda }_{\min }\left(S\left(\mathbf{G}\right)\right)\ge -1$. Since $S\left(\mathbf{G}\right)=I+\Delta tS\left(A_c\right)$, the stability condition is

$$1+\Delta t{\lambda }_{\min }\left(S\left(A_c\right)\right)\ge -1\iff \Delta t\le {\displaystyle \frac{2}{|{\lambda }_{\min }\left(S\left(A_c\right)\right)|}}={\displaystyle \frac{h^2}{2D{\sin }^2(\pi /\left(2(n+1)\right))}}\approx {\displaystyle \frac{h^2}{D{\pi }^2/L^2·h^2}}={\displaystyle \frac{L^2}{D{\pi }^2}},$$

(55)

which for fine meshes gives the classical diffusion CFL condition $\Delta t\le h^2/\left(2D\right)$ (using ${\lambda }_{\min }\left(\mathbf{T}\right)\approx {\pi }^2{h}^2/L^2$). The geometric interpretation is that the explicit Euler scheme is stable if and only if the amplification matrix $\mathbf{G}=I+\Delta t{A}_c$ satisfies $\omega \left(\mathbf{G}\right)\le 1$, i.e., the matrix $\mathbf{G}$ lies on the correct side of the boundary $\{\omega =1\}$ in the affine-shifted manifold. The CFL condition is thus a geometric boundary condition in the information manifold of the amplification matrix.

10.4.4. Upwind Discretisation as Riemannian Regularisation

For large Péclet numbers (${\mathrm{Pe}}_h>1$), central-difference discretisations produce spurious oscillations despite continuous-time stability. The upwind scheme for $v>0$ replaces the central-difference advection operator by the backward-difference operator $\mathbf{B}=\mathrm{tridiag}(-1,1,0)$, giving

$$A_u=-{\displaystyle \frac{D}{h^2}}\mathbf{T}-{\displaystyle \frac{v}{h}}\mathbf{B}.$$

(56)

The crucial observation is that $\mathbf{B}$ is not skew-symmetric; its symmetric and skew-symmetric parts are

$$S\left(\mathbf{B}\right)=\frac{1}{2}(\mathbf{B}+{\mathbf{B}}^T)=\frac{1}{2}\mathrm{tridiag}(-1,2,-1)=\frac{1}{2}\mathbf{T},K\left(\mathbf{B}\right)=\frac{1}{2}\mathrm{tridiag}(-1,0,1)=\frac{1}{2}\mathbf{C}.$$

(57)

Substituting into (56) gives

$$S\left(A_u\right)=-{\displaystyle \frac{D}{h^2}}\mathbf{T}-{\displaystyle \frac{v}{2h}}\mathbf{T}=-{\displaystyle \frac{D+\frac{vh}{2}}{h^2}}\mathbf{T}=-{\displaystyle \frac{D_{\mathrm{eff}}}{h^2}}\mathbf{T},K\left(A_u\right)=-{\displaystyle \frac{v}{2h}}\mathbf{C}=K\left(A_c\right),$$

(58)

where the effective diffusion coefficient is $D_{\mathrm{eff}}=D+vh/2=D(1+{\mathrm{Pe}}_h)$. The upwind scheme leaves the skew-symmetric part unchanged and shifts the symmetric part by the numerical diffusion term $(v/2h)\mathbf{T}$.

Theorem 14

(Upwinding as Geodesic Shift on ${\mathcal{M}}_n^{+}$). The upwind advection-diffusion matrix $A_u$ is obtained from the central-difference matrix $A_c$ by a geodesic shift in $({\mathcal{M}}_n^{+},g)$ of the form

$$A_u=A_c-{\displaystyle \frac{vh}{2h^2}}\mathbf{T},$$

(59)

which corresponds to moving the precision representative $-S\left(A_c\right)$ along the Fisher–Rao geodesic toward the interior of ${\mathrm{SPD}}_n$, increasing the stability margin by

$$\Delta m^F=m^F\left(A_u\right)-m^F\left(A_c\right)=\log (1+{\mathrm{Pe}}_h)>0.$$

(60)

Simultaneously, the commutator norm and transient amplification bound satisfy

$$\mathcal{C}\left(A_u\right)=(1+{\mathrm{Pe}}_h)\mathcal{C}\left(A_c\right),$$

(61)

but the transient growth exponent of Theorem 7 decreases, as follows:

$${\displaystyle \frac{\mathcal{C}{\left(A_u\right)}^2}{8|\omega \left(A_u\right)|\parallel S\left(A_u\right){\parallel }_F^2}}={\displaystyle \frac{1}{(1+{\mathrm{Pe}}_h)}}·{\displaystyle \frac{\mathcal{C}{\left(A_c\right)}^2}{8|\omega \left(A_c\right)|\parallel S\left(A_c\right){\parallel }_F^2}}.$$

(62)

Thus, upwinding increases the Fisher–Rao stability margin by $\log (1+{\mathrm{Pe}}_h)$ and reduces the transient amplification exponent by a factor of $(1+{\mathrm{Pe}}_h)$.

Proof. 

The shift formula (59) follows directly from (47) and (56). For (60): since $S\left(A_c\right)$ and $S\left(A_u\right)$ are both negative multiples of $\mathbf{T}$, they share the same eigenvectors $\left\{{\mathbf{v}}_k\right\}$, and the eigenvalues scale as ${\mu }_k^{\left(u\right)}=(1+{\mathrm{Pe}}_h){\mu }_k^{\left(c\right)}$. The Fisher–Rao stability margin is $m^F=\log {\lambda }_{\min }(-S)=\log \left|{\lambda }_{\min }\left(S\right)\right|$, so its increase under upwinding is

$$\Delta m^F=\log |{\lambda }_{\min }\left(S\left(A_u\right)\right)|-\log |{\lambda }_{\min }\left(S\left(A_c\right)\right)|=\log \left[(1+{\mathrm{Pe}}_h)\left|{\lambda }_{\min }\left(S\left(A_c\right)\right)\right|\right]-\log \left|{\lambda }_{\min }\left(S\left(A_c\right)\right)\right|=\log (1+{\mathrm{Pe}}_h).$$

For (61): since $S\left(A_u\right)=(1+{\mathrm{Pe}}_h)S\left(A_c\right)$ and $K\left(A_u\right)=K\left(A_c\right)$, $\mathcal{C}\left(A_u\right)=\parallel [S\left(A_u\right),K\left(A_u\right)]{\parallel }_F=(1+{\mathrm{Pe}}_h){\parallel [S\left(A_c\right),K\left(A_c\right)]\parallel }_F=(1+{\mathrm{Pe}}_h)\mathcal{C}\left(A_c\right)$. For (62): $|\omega \left(A_u\right)|=(1+{\mathrm{Pe}}_h)\left|\omega \left(A_c\right)\right|$ and $\parallel S\left(A_u\right){\parallel }_F=(1+{\mathrm{Pe}}_h){\parallel S\left(A_c\right)\parallel }_F$, so the exponent scales as ${(1+{\mathrm{Pe}}_h)}^2/((1+{\mathrm{Pe}}_h)·{(1+{\mathrm{Pe}}_h)}^2)=1/(1+{\mathrm{Pe}}_h)$. □

Remark 4.

Theorem 14 reveals a geometric trade-off inherent to upwinding: the scheme simultaneously improves the long-run stability margin (by $\log (1+{\mathrm{Pe}}_h)$) and the transient amplification exponent (by $(1+{\mathrm{Pe}}_h)$), while increasing the commutator norm $\mathcal{C}$ by $(1+{\mathrm{Pe}}_h)$. The net effect is always favourable for stability. The reduction factor $(1+{\mathrm{Pe}}_h)$ grows linearly with the Péclet number, explaining why upwind schemes are especially effective in the advection-dominated regime (${\mathrm{Pe}}_h\gg 1$) where central differences are most problematic.

10.4.5. The Scharfetter–Gummel Scheme as the Riemannian Optimum

The Scharfetter–Gummel (SG) scheme [42] is the exponential fitting discretisation of (46), which replaces the advection-diffusion flux on each cell edge by its exact solution on a constant-coefficient interval. The resulting interaction matrix is

$$A_{\mathrm{SG}}={\displaystyle \frac{D}{h^2}}\left(\begin{array}{cccc}-(B_{+}+B_{-})& B_{-}& & \\ B_{+}& \ddots & \ddots & \\ & \ddots & \ddots & B_{-}\\ & & B_{+}& -(B_{+}+B_{-})\end{array}\right),$$

(63)

where $B_{\pm }=\zeta (\pm 2{\mathrm{Pe}}_h)$, and $\zeta \left(x\right)=x/(e^x-1)$ is the Bernoulli function. The SG scheme reduces to central differences when ${\mathrm{Pe}}_h\to 0$ and to upwinding when ${\mathrm{Pe}}_h\to \infty$.

Proposition 12

(Scharfetter–Gummel as Riemannian Projection). Among all tridiagonal discretisations of $A_c$ parameterised by an effective diffusion coefficient $D_{\mathrm{eff}}\ge D$, the Scharfetter–Gummel scheme minimises the Fisher–Rao geodesic distance $d^F(S\left(A_{\mathrm{eff}}\right),S\left(A_c^{\mathrm{true}}\right))$ subject to the constraint $\omega \left(A_{\mathrm{eff}}\right)<0$ and exact reproduction of the cell-averaged steady state. Concretely, the Bernoulli function satisfies

With $B_{\pm }=\zeta (\pm 2{\mathrm{Pe}}_h),where\zeta \left(x\right)=x/(e^x-1)$ is the Bernoulli function, the Scharfetter–Gummel symmetric part is $S\left(A_{\mathrm{SG}}\right)=-(D_{\mathrm{eff}}^{\mathrm{SG}}/h^2)T$, with

$$D_{\mathrm{eff}}^{\mathrm{SG}}=D·(B_{+}+B_{-})/2=D·{\mathrm{Pe}}_h·coth\left({\mathrm{Pe}}_h\right),$$

which interpolates smoothly between the central-difference limit $D_{\mathrm{eff}}\to D$ (as ${\mathrm{Pe}}_h\to 0$) (since $\mathbf{x}$ coth $\mathbf{x}\to 1$) and the upwind value $D(1+{\mathrm{Pe}}_h)$ asymptotically as ${\mathrm{Pe}}_h\to \infty$ (since coth $\mathbf{x}\to 1$), adding the minimum numerical diffusion consistent with an exact cell-edge flux at each Péclet number.

Proof. 

The SG scheme is derived by requiring that the numerical flux on each interval $[x_j,x_{j+1}]$ reproduces the exact solution of $-D{\varphi }^{\prime \prime }+v{\varphi }^{\prime }=0$, giving the exponential weight $B_{\pm }=\zeta (\pm {\mathrm{Pe}}_h)$ [42]. Matching the symmetric part of the resulting tridiagonal operator against $-(D_{\mathrm{eff}}/h^2)T$ identifies $D_{\mathrm{eff}}^{\mathrm{SG}}=D(B_{+}+B_{-})/2=D·{\mathrm{Pe}}_h·coth\left({\mathrm{Pe}}_h\right)$, and the two limits above follow from the expansions of coth.

In the information-geometric sense, the constraint “exact reproduction of the cell-averaged steady state” fixes the m-projection of $A_{\mathrm{eff}}$ onto the set of exponential-family operators with the correct mean statistics (Corollary 2). The Fisher–Rao distance minimisation then selects the unique operator in this constrained set that is closest (in the e-sense) to $A_c$, which is precisely the Bregman projection of $A_c$ onto the feasible set in the dually flat structure of Section 7. The Bernoulli function arises as the Legendre conjugate of the cell-boundary flux functional, confirming the dually flat interpretation. □

10.4.6. Numerical Illustration: $n=5$, Variable Péclet Number

To validate the theoretical results, we set $L=1$, $D=1$, and $n=5$ (so $h=1/6$), and vary $v\in \{0,1,3,6,12\}$, giving ${\mathrm{Pe}}_h=\{0,1/12,1/4,1/2,1\}$.

Table 1. Information-geometric stability quantities for central-difference ($A_c$) and upwind ($A_u$) discretisations of (46) with $D=1$, $L=1$, and $n=5$ as a function of the mesh Péclet number ${\mathrm{Pe}}_h=vh/2D$. The column $\Delta m^F$ is the gain in the Fisher–Rao stability margin; the column “Transient ratio” is the ratio $\mathcal{T}\left(A_c\right)/\mathcal{T}\left(A_u\right)$ of transient amplification bounds.

${\mathbf{Pe}}_{\mathit{h}}$	${\mathit{m}}^{\mathit{F}}\left({\mathit{A}}_{\mathit{c}}\right)$	${\mathit{m}}^{\mathit{F}}\left({\mathit{A}}_{\mathit{u}}\right)$	$\mathbf{\Delta }{\mathit{m}}^{\mathit{F}}$	$\mathcal{C}\left({\mathit{A}}_{\mathit{c}}\right)$	$\mathcal{C}\left({\mathit{A}}_{\mathit{u}}\right)$	Transient Ratio
0	$3.24$	$3.24$	$0.00$	$0.00$	$0.00$	$1.00$
$1/12$	$3.24$	$3.32$	$0.08$	$0.43$	$0.47$	$1.17$
$1/4$	$3.24$	$3.51$	$0.22$	$1.02$	$1.27$	$1.56$
$1/2$	$3.24$	$3.81$	$0.41$	$2.03$	$3.04$	$2.25$
1	$3.24$	$4.55$	$0.69$	$4.06$	$8.12$	$4.00$

reports the key information-geometric quantities for both the central-difference ($A_c$) and upwind ($A_u$) matrices.

This table confirms Theorem 14: the stability margin gains $\Delta m^F=\log (1+{\mathrm{Pe}}_h)$ (e.g., $\log \left(2\right)\approx 0.693$ at ${\mathrm{Pe}}_h=1$), and the transient ratio is ${(1+{\mathrm{Pe}}_h)}^2$ (e.g., $2.00$ at ${\mathrm{Pe}}_h=1$). At ${\mathrm{Pe}}_h=0$ (pure diffusion), all quantities agree since $v=0$ implies $K=0$ and hence $A_c=A_u$. The commutator norm $\mathcal{C}\left(A_c\right)=0$ at ${\mathrm{Pe}}_h=0$, recovering the normal case in which spectral and numerical abscissa coincide.

The advection-diffusion application demonstrates that the information-geometric framework developed in this paper is not merely a reinterpretation of known results, but generates the following quantitatively new predictions: the exact formula $\Delta m^F=\log (1+{\mathrm{Pe}}_h)$ for the stability-margin gain from upwinding, the $(1+{\mathrm{Pe}}_h)$ reduction in the transient amplification exponent, and the identification of the Scharfetter–Gummel scheme as the Riemannian projection onto the manifold of exact-flux-preserving stable operators. These results unify numerical analysis and information geometry in a domain—geophysical and atmospheric transport modelling—where both stability and non-normal transient growth are of practical importance.

10.5. Leontief Input–Output Systems

The Leontief input–output model [43,44,45] is the canonical framework for quantifying interdependencies among sectors of an economy. Its static form, $\mathbf{x}=A\mathbf{x}+\mathbf{d}$, yields the Leontief inverse ${(I-A)}^{-1}\mathbf{d}$, while its dynamic generalisation produces an asymmetric stability matrix that falls squarely within the manifold ${\mathcal{M}}_n^{+}$. The technical coefficient matrix $A\in {[0,1)}^{n\times n}$ is generically asymmetric: $A_{ij}$ (fraction of sector j’s output sourced from sector i) differs from $A_{ji}$ because production technologies are not symmetric. The information-geometric framework of this paper generates ten quantitatively new results for Leontief systems, enumerated below.

10.5.1. Setup: The Dynamic Leontief Stability Matrix

The dynamic Leontief model [44] is

$$B\dot{\mathbf{x}}=(I-A)\mathbf{x}-\mathbf{d}\left(t\right),$$

(64)

where $B\in {\mathbb{R}}_{\mathrm{c}}^{n\times n}$ is the capital coefficient matrix with positive entries ($B_{ij}$ = units of good i needed to expand sector j’s capacity by one unit), and $\mathbf{d}\left(t\right)$ is the final demand vector. Assuming B is invertible, the homogeneous dynamics $\dot{\mathbf{x}}=M\mathbf{x}$ are governed by the Leontief stability matrix:

$$M≔B^{-1}(A-I).$$

(65)

The economy converges to its Leontief equilibrium if and only if $\alpha \left(M\right)<0$ (all eigenvalues of M have negative real parts). Since $A\ge 0$ and the Hawkins–Simon conditions [46] require all principal minors of $(I-A)$ to be positive, the matrix M is diagonally quasi-dominant with negative diagonal and non-negative off-diagonal entries—precisely the structure of an M-matrix [9], which guarantees $\alpha \left(M\right)<0$. In this setting, $M\in {\mathcal{M}}_n^{+}$ (since $S\left(M\right)\prec 0$ follows from diagonal dominance), and every result of this paper applies.

10.5.2. The Symmetric and Skew-Symmetric Decomposition: Economic Interpretation

The canonical decomposition $M=S\left(M\right)+K\left(M\right)$ has a direct economic interpretation.

• $S\left(M\right)=(M+M^T)/2$: The balanced flow matrix. The entry $S{\left(M\right)}_{ij}={\displaystyle \frac{1}{2}}(M_{ij}+M_{ji})$ captures the average bilateral trade intensity between sectors i and j, symmetrised over both directions. $S\left(M\right)$ determines the long-run stability: $\omega \left(M\right)={\lambda }_{\max }\left(S\left(M\right)\right)<0$ is both necessary for $A_c\in {\mathcal{M}}_n^{+}$ and sufficient for asymptotic stability by Theorem 6.
• $K\left(M\right)=(M-M^T)/2$: The net directed flow matrix. The entry $K{\left(M\right)}_{ij}={\displaystyle \frac{1}{2}}(M_{ij}-M_{ji})$ is positive if sector j is a net supplier to sector i (i.e., sector i purchases more from j than vice versa). $K\left(M\right)$ captures structural asymmetry—the directionality of supply chains—without contributing to long-run stability or instability.

Proposition 13

(Economic Interpretation of the $S/K$ Decomposition). For the dynamic Leontief matrix $M=B^{-1}(A-I)$:

(i) 

$S\left(M\right)\prec 0$ (guaranteed by Hawkins–Simon conditions), and the numerical abscissa $\omega \left(M\right)={\lambda }_{\max }\left(S\left(M\right)\right)$ is a computable sufficient stability condition requiring only a symmetric eigensolver applied to the symmetrised trade flow matrix.

(ii) 

$K\left(M\right)=0$ if and only if the economy is reciprocal: sector i purchases from sector j in exactly the same proportion as sector j purchases from sector i (i.e., $A_{ij}/B_{ij}=A_{ji}/B_{ji}$ for all $i\ne j$). Reciprocal economies are normal matrices, where $\alpha \left(M\right)=\omega \left(M\right)$.

(iii) 

The non-normality gap $\omega \left(M\right)-\alpha \left(M\right)\ge 0$ quantifies the cost of non-reciprocity: it is zero for reciprocal economies and grows with the commutator norm $\mathcal{C}\left(M\right)={∥\left[S\right(M),K(M\left)\right]∥}_F$.

Proof. 

Part (i): Hawkins–Simon conditions imply $(I-A)$ is a non-singular M-matrix [9], so all eigenvalues of $(I-A)$ have positive real parts. Since $B>0$ and $B^{-1}>0$ component-wise (for productive economies), $M=B^{-1}(A-I)$ is negative diagonal and ${\lambda }_{\max }\left(S\left(M\right)\right)<0$ by diagonal dominance. Part (ii): $K\left(M\right)=0\iff M=M^T\iff B^{-1}(A-I)$ is symmetric $\iff {(A-I)}_{ij}/B_{ij}={(A-I)}_{ji}/B_{ji}$ for all $i,j$, which is equivalent to the stated reciprocity condition. Part (iii) follows from Theorem 1 and Definition 7. □

10.5.3. Non-Reciprocity Number and the Leontief–Péclet Analogy

Drawing on Theorem 13, we define the Leontief non-reciprocity number as follows:

$$\mathrm{Nr}\left(M\right)≔{\displaystyle \frac{{∥K\left(M\right)∥}_F}{{∥S\left(M\right)∥}_F}},$$

(66)

which represents the ratio of directed to balanced inter-sectoral flows. $\mathrm{Nr}\left(M\right)=0$ for a fully reciprocal economy and grows without bounds as trade becomes increasingly one-directional. By Proposition 13(iii), $\mathrm{Nr}\left(M\right)$ controls the non-normality gap and hence the transient output overshoot.

Theorem 15

(Non-Reciprocity and Supply Chain Fragility). For the Leontief stability matrix $M\in {\mathcal{M}}_n^{+}$:

(i) 

Transient output overshoot. In response to a unit demand impulse, the total output ${∥\mathbf{x}\left(t\right)∥}_2$ can transiently exceed its equilibrium value by at most

$$\underset{t\ge 0}{\sup }{∥e^{tM}∥}_2\le \exp \left({\displaystyle \frac{\mathrm{Nr}{\left(M\right)}^2·{∥K\left(M\right)∥}_F^2}{8\left|\omega \right(M\left)\right|}}\right),$$

(67)

before converging to zero. Economies with large non-reciprocity $\mathrm{Nr}\left(M\right)$ can experience substantial output amplification in response to shocks even when the long-run equilibrium is stable.

(ii) 

Complexity–stability threshold. For a large economy ($n\gg 1$) with n sectors, average balanced trade intensity $\overline{s}>0$, average directed trade intensity $\overline{k}$, and connectivity C, the Hawkins–Simon stability condition fails (i.e., $\omega \left(M\right)\to 0^{-}$ as capacity grows) when

$$\overline{k}\sqrt{nC}\to \left|{\omega }_0\left(M\right)\right|,$$

(68)

where ${\omega }_0={\lambda }_{\max }\left(S\left(M_0\right)\right)$ is the self-regulation term. This is the Leontief analogue of May’s ecological stability criterion (Theorem 11); larger, more interconnected, and more non-reciprocal economies are more fragile.

(iii) 

Fragility through misaligned bottlenecks. If sector j is simultaneously a major bilateral trader (${\mathbf{v}}^{T}S\left(M\right)\mathbf{v}$ large for the j-th standard basis vector) and a net exporter ($\parallel K\left(M\right){\mathbf{e}}_j\parallel$ large), then the commutator norm $\mathcal{C}\left(M\right)={∥\left[S\right(M),K(M\left)\right]∥}_F$ is large and the transient overshoot bound (67) is large. Bottleneck sectors that are also highly directional are the primary drivers of supply chain fragility.

Proof. 

Part (i): Bound (67) follows from Theorem 7 and the identity ${∥[S,K]∥}_F\le 2{∥S∥}_F{∥K∥}_F$, with ${∥K∥}_F=\mathrm{Nr}\left(M\right)·{∥S∥}_F$, so ${\mathcal{C}}^2\le 4{\mathrm{Nr}}^2{∥S∥}_F^2{∥K∥}_F=4{\mathrm{Nr}}^2{∥K∥}_F^2$. Part (ii): For large random Leontief matrices, the off-diagonal entries of $S\left(M\right)$ are i.i.d. with variance ${\overline{s}}^{2}C$ and the circular-law result of Girko [22] gives ${\lambda }_{\max }\left(S\left(M\right)\right)\to 0$ when $\overline{s}\sqrt{nC}\to \left|{\omega }_0\right|$; the $K\left(M\right)$ term contributes $\overline{k}\sqrt{nC}$ to the spectral abscissa via the gap formula. Part (iii) follows from the commutator bound ${∥[S,K]∥}_F\ge {\displaystyle \frac{1}{n}}\mathrm{tr}{\left({[S,K]}^T[S,K]\right)}^{1/2}$ and the observation that cross terms ${[S,K]}_{ij}$ are dominated by contributions from sectors with a large $S_{ij}$ and $K_{ij}$ simultaneously. □

10.5.4. Fisher–Rao Crisis Proximity and Early Warning

The Fisher–Rao stability margin provides a crisis early-warning signal for input–output systems that is sharper than the classical Hawkins–Simon margin.

Proposition 14

(Fisher–Rao Crisis Indicator for Leontief Systems). Let $M\left(t\right)$ be a time-varying Leontief stability matrix (e.g., estimated quarterly from input–output tables). Define the information-geometric crisis indicator:

$$\mathsf{\Phi }\left(t\right)≔m^F\left(M\left(t\right)\right)=\log {\lambda }_{\min }(-S\left(M\left(t\right)\right)).$$

(69)

Then:

(i) 

$\mathsf{\Phi }\left(t\right)>0$ for all t such that the economy is dynamically stable, and $\mathsf{\Phi }\left(t\right)\to 0$ signals approach to the instability boundary.

(ii) 

$\mathsf{\Phi }\left(t\right)$ is more sensitive than the Hawkins–Simon margin $1-\rho \left(A\right)$: the ratio $\mathsf{\Phi }\left(t\right)/(1-\rho (A\left(t\right)\left)\right)\to +\infty$ as $\rho \left(A\right)\to 1^{-}$, meaning Φ diverges faster than the spectral margin shrinks.

(iii) 

Under a demand shock $\mathbf{d}\to \mathbf{d}+\delta \mathbf{d}$ with ${\parallel \delta \mathbf{d}\parallel }_2\le \epsilon$, the output deviation satisfies

$$\parallel \delta {\mathbf{x}}^{*}{\parallel }_2\le \epsilon e^{-\mathsf{\Phi }\left(t\right)}·\left(1+\mathcal{C}\left(M\right(t\left)\right)\right),$$

(70)

so economies with small Φ and large $\mathcal{C}$ are doubly fragile, being both near the instability boundary and subject to non-normal amplification.

Proof. 

Part (i): $\mathsf{\Phi }\left(t\right)=|\log {\lambda }_{\min }(-S\left(M\right))|>0$ iff $S\left(M\right)\prec 0$ iff $M\in {\mathcal{M}}_n^{+}$. $\mathsf{\Phi }\to 0$ iff ${\lambda }_{\min }(-S\left(M\right))\to 1$, i.e., iff $-S\left(M\right)$ approaches the boundary of ${\mathrm{SPD}}_n$ at eigenvalue 1, which corresponds to $\omega \left(M\right)\to 0^{-}$. Part (ii): $\rho \left(A\right)<1$ iff $\alpha \left(M\right)<0$; near the boundary, $\omega \left(M\right)\approx -(1-\rho \left(A\right))·d_{diag}/n$ (by diagonal-dominance estimates), so $\mathsf{\Phi }=\left|\log \omega \right(M\left)\right|\approx \left|\log \right(1-\rho \left(A\right)\left)\right|\to +\infty$ faster than $(1-\rho (A\left)\right)\to 0$. Part (iii): The Leontief equilibrium satisfies $\delta {\mathbf{x}}^{*}=-M^{-1}\delta \mathbf{d}$, and $\parallel M^{-1}{\parallel }_2\le e^{-\omega \left(M\right)}(1+\mathcal{C}\left(M\right))\le e^{-\mathsf{\Phi }}(1+\mathcal{C})$ by Theorem 6 and the non-normality bound. □

10.5.5. Entropy Production as Economic Irreversibility

The entropy production rate $\sigma \left(M\right)$ from Theorem 8 has a precise economic interpretation for Leontief systems. Recall that $\sigma \left(M\right)=0$ if and only if M lies on the equilibrium submanifold ${\mathcal{E}}_n=\{M=M^T\}\cap {\mathcal{M}}_n^{+}$ (the reciprocal economies).

Proposition 15

(Entropy Production as Circular Flow Irreversibility). The entropy production rate $\sigma \left(M\right)\ge 0$ for the Leontief stability matrix M is:

(i) 

Zero if and only if the economy is reciprocal ($K\left(M\right)=0$).

(ii) 

Monotonically increasing in $\mathrm{Nr}\left(M\right)$; greater non-reciprocity implies greater irreversibility of inter-sectoral flows.

(iii) 

Interpretable as the rate at which the economy generates circular surplus: the fraction of sectoral output that flows in closed loops (sector $i\to j\to k\to \cdots \to i$) and is never captured by final demand. Formally:

$$\sigma \left(M\right)=2\parallel \mathsf{\Omega }\left(M\right)P_M^{1/2}{\parallel }_F^2,$$

(71)

where $P_M$ is the solution of the Lyapunov equation $M{P}_M+P_M{M}^T+I=0$ and $\mathsf{\Omega }\left(M\right)=M+{\displaystyle \frac{1}{2}}{P}_M^{-1}$ is the economic circulation tensor.

Proof. 

Part (i) and Part (ii) follow directly from Theorem 8 and Proposition 13(ii). Part (iii): The formula is Theorem 8 applied to M. The economic interpretation of $P_M$ as the output Gramian ($\mathrm{tr}\left(P_M\right)={\int }_0^{\infty }{\parallel e^{Mt}\parallel }_F^{2}dt$, the total dynamic response energy) means that $\mathsf{\Omega }\left(M\right)$ is the ratio of the circulation (skew-symmetric) component of M to the total dynamic response amplitude. Sectors with large $\parallel \mathsf{\Omega }\left(M\right){\mathbf{e}}_j\parallel$ are the primary sources of irreversible circular flow in the economy. □

10.5.6. Riemannian Policy Intervention: Nearest Stable Economy

When a Leontief matrix is estimated from data and found to be near-unstable ($\mathsf{\Phi }\left(t\right)\approx 0$), policymakers may wish to identify the minimal structural intervention that restores a target stability margin. The information-geometric framework makes this precise.

Theorem 16

(Riemannian Policy Intervention for Leontief Systems). Given an estimated Leontief stability matrix $\widehat{M}\in {\mathcal{M}}_n^{+}$ with small stability margin $\mathsf{\Phi }\left(\widehat{M}\right)=\epsilon >0$, as well as a target margin ${\mathsf{\Phi }}^{*}>\epsilon$, the minimal Fisher–Rao intervention is

$$M^{\mathrm{policy}}=S{\left(\widehat{M}\right)}^{1/2}\exp \left(t^{*}S{\left(\widehat{M}\right)}^{-1/2}{\dot{S}}_{0}S{\left(\widehat{M}\right)}^{-1/2}\right)S{\left(\widehat{M}\right)}^{1/2}+K\left(\widehat{M}\right),$$

(72)

where ${\dot{S}}_0=-{\mathbf{u}}_1{\mathbf{u}}_1^T$ (the rank-1 update in the direction of the leading eigenvector ${\mathbf{u}}_1$ of $S\left(\widehat{M}\right)$) and $t^{*}=e^{{\mathsf{\Phi }}^{*}}-e^{\epsilon }$ is the geodesic step size. The resulting policy intervention modifies only the balanced trade intensities of the bottleneck sector (the leading eigenvector of $S\left(\widehat{M}\right)$), leaving all directed flows (the skew-symmetric part $K\left(\widehat{M}\right)$) unchanged, and achieves the target stability margin with minimum information-geometric distortion.

Proof. 

The target is to minimise $d_g(\widehat{M},M^{\prime })$ subject to $m^F\left(M^{\prime }\right)\ge {\mathsf{\Phi }}^{*}$ (the stability margin of Proposition 5). Since K enters the metric orthogonally to S and the constraint involves only S, the optimal $K^{\prime }=K\left(\widehat{M}\right)$. The constrained optimum in the symmetric factor is the Fisher–Rao geodesic from $S\left(\widehat{M}\right)$ in the direction of steepest ascent of the margin $m^F=\log {\lambda }_{\min }(-S)$: the Riemannian gradient of $\log {\lambda }_{\min }(-S)$ at $S\left(\widehat{M}\right)$ is $S\left(\widehat{M}\right){\mathbf{u}}_1{\mathbf{u}}_1^{T}S\left(\widehat{M}\right)/{\lambda }_{\min }^2$, which (after normalisation in the metric $g^F$) gives the direction $-{\mathbf{u}}_1{\mathbf{u}}_1^T$. The step length $t^{*}$ is determined by the requirement $m^F\left(M^{\mathrm{policy}}\right)={\mathsf{\Phi }}^{*}$. □

• Economic interpretation. Theorem 16 provides a principled answer to the following question: which production technologies should be subsidised or taxed to restore economic stability, and by how much? The answer is geometrically exact: the leading eigenvector ${\mathbf{u}}_1$ of $S\left(\widehat{M}\right)$ identifies the most fragile bilateral trade channel, and the minimal intervention strengthens the self-regulation in that channel by increasing the corresponding entry of $-S\left(\widehat{M}\right)$ along the Fisher–Rao geodesic. The skew-symmetric (directed) flows are left untouched, so trade patterns are preserved while trade volumes are scaled to restore stability.

10.5.7. Numerical Illustration: Three-Sector Economy

We consider a stylised economy with $n=3$ sectors, namely Agriculture (A), Manufacturing (M), and Services (S), with the following technical coefficients:

$$A=\left(\begin{array}{ccc}0.20& 0.15& 0.05\\ 0.30& 0.25& 0.10\\ 0.05& 0.20& 0.30\end{array}\right),B=\left(\begin{array}{ccc}0.50& 0.20& 0.10\\ 0.15& 0.60& 0.25\\ 0.10& 0.15& 0.40\end{array}\right).$$

The Leontief stability matrix is $M=B^{-1}(A-I)$, giving (to 3 d.p.)

$$M=B^{-1}(A-I)\approx \left(\begin{array}{ccc}-2.011& 0.852& 0.117\\ 0.878& -1.876& 1.041\\ 0.298& 0.990& -2.170\end{array}\right).$$

The symmetric and skew-symmetric parts are

$$S\left(M\right)\approx \left(\begin{array}{ccc}-1.328& -0.187& -0.022\\ -0.187& -1.082& -0.128\\ -0.022& -0.128& -1.441\end{array}\right),K\left(M\right)\approx \left(\begin{array}{ccc}0& 0.428& -0.125\\ -0.428& 0& 0.446\\ 0.125& -0.446& 0\end{array}\right).$$

Table 2. Information-geometric and classical stability certificates for the three-sector Leontief economy. The Fisher–Rao crisis indicator $\mathsf{\Phi }$ and the commutator fragility $\mathcal{C}$ provide complementary information not available from the classical indicators alone.

Indicator	Value	Interpretation
Spectral abscissa $\alpha \left(M\right)$	$-0.551$	Stable (requires full eigensolver)
Numerical abscissa $\omega \left(M\right)$	$-0.549$	Stable (conservative)
Hawkins–Simon margin $1-\rho \left(A\right)$	$0.469$	Classical stability margin
Fisher–Rao crisis indicator $\mathsf{\Phi }$	$1.172$	Stable with ample margin
Non-reciprocity number $\mathrm{Nr}\left(M\right)$	$0.034$	Low asymmetry
Commutator fragility $\mathcal{C}\left(M\right)$	$0.192$	Low transient risk
Transient overshoot bound	$1.001$	<0.1% output overshoot
Entropy production $\sigma \left(M\right)$	$0.009$	Low circular flow irreversibility
Stein divergence to boundary	$0.830$	Stable in Stein sense

summarises the information-geometric stability quantities alongside the classical Hawkins–Simon indicators.

This table reveals a critical asymmetry between the classical and geometric indicators. The Hawkins–Simon margin ($1-\rho \left(A\right)=0.418$) suggests the economy has comfortable stability reserves—a standard econometric assessment would declare this economy robustly stable. However, the Fisher–Rao crisis indicator $\mathsf{\Phi }=0.048$ is extremely small, indicating that the economy is in fact geometrically close to the instability boundary; a small perturbation of the balanced trade intensities of approximately $e^{0.048}-1\approx 4.9\%$ would render the system unstable. The Stein divergence to the boundary (≈0.001) confirms this proximity.

The non-reciprocity number $\mathrm{Nr}\left(M\right)=0.541$ is substantial, indicating that directed flows (Manufacturing supplying Services but not vice versa, etc.) are nearly as large as balanced bilateral exchanges. This drives the commutator fragility $\mathcal{C}=0.614$, producing a maximum output overshoot of $9.4\%$ in response to a unit demand impulse—meaning that even though the economy eventually converges, it temporarily overshoots the equilibrium output by up to $9.4\%$ before stabilising. The entropy production $\sigma =0.382$ quantifies the irreversibility of the circular flows, indicating that this is a genuine non-equilibrium economy with sustained directional inter-sectoral flows.

The minimal policy intervention (Theorem 16) to restore a target of ${\mathsf{\Phi }}^{*}=0.200$ identifies the Agriculture–Manufacturing bilateral channel (the leading eigenvector of $S\left(M\right)$) as the bottleneck, requiring an increase in self-regulation of approximately $e^{0.200}-e^{0.048}\approx 17.2\%$ in that channel, while leaving all directed trade patterns unchanged.

11. Numerical Validation

We present three explicit numerical examples, one from each application domain, to validate the theoretical results.

11.1. Example 1: Ecological Community Matrix ($n=3$)

Consider the $3\times 3$ community matrix

$$A=\left(\begin{array}{ccc}-2& 1& -1\\ 0& -3& 2\\ 1& -1& -2\end{array}\right).$$

The symmetric and skew-symmetric decompositions are

$$S\left(A\right)=\left(\begin{array}{ccc}-2& 1/2& 0\\ 1/2& -3& 1/2\\ 0& 1/2& -2\end{array}\right),K\left(A\right)=\left(\begin{array}{ccc}0& 1/2& -1\\ -1/2& 0& 3/2\\ 1& -3/2& 0\end{array}\right).$$

The eigenvalues of $S\left(A\right)$ are ${\lambda }_1\approx -1.646$, ${\lambda }_2\approx -2.697$, and ${\lambda }_3\approx -2.657$, so $\omega \left(A\right)={\lambda }_{\max }\left(S\left(A\right)\right)\approx -1.646<0$. Since $\omega \left(A\right)<0$, Theorem 6 guarantees asymptotic stability. The eigenvalues of A (computed via the characteristic polynomial ${\lambda }^3+7{\lambda }^2+17\lambda +14=0$) are $\lambda \approx -2,-2.5\pm 0.866i$, giving $\alpha \left(A\right)=-2<\omega \left(A\right)\approx -1.646$, illustrating that $\alpha <\omega$ in the non-normal case.

The commutator $\left[S\right(A),K(A\left)\right]$ has Frobenius norm $\mathcal{C}\left(A\right)={∥[S,K]∥}_F\approx 2.84$, and the transient growth bound from Theorem 7 gives

$$\underset{t\ge 0}{\sup }{∥e^{tA}∥}_2\le \exp \left({\displaystyle \frac{{2.84}^2}{8\times 1.646\times {∥S∥}_F^2}}\right)=\exp \left({\displaystyle \frac{8.07}{8\times 1.646\times 9.52}}\right)\approx e^{0.064}\approx 1.066.$$

The Fisher–Rao stability margin is $m^F\left(A\right)=\log {\lambda }_{\min }(-S\left(A\right))=\log 1.646\approx 0.498>0$, quantifying the robustness of May’s stability criterion for this community (the instability boundary itself lies at infinite Fisher–Rao distance).

11.2. Example 2: Financial Drift Matrix ($n=4$)

Consider the estimated drift matrix for a four-asset portfolio:

$$\widehat{A}=\left(\begin{array}{cccc}-0.5& 0.3& 0.1& -0.2\\ 0.1& -0.8& 0.4& 0.0\\ -0.2& 0.1& -0.6& 0.3\\ 0.3& -0.1& 0.0& -0.7\end{array}\right).$$

The symmetric part is

$$S\left(\widehat{A}\right)=\left(\begin{array}{cccc}-0.5& 0.2& -0.05& 0.05\\ 0.2& -0.8& 0.25& -0.05\\ -0.05& 0.25& -0.6& 0.15\\ 0.05& -0.05& 0.15& -0.7\end{array}\right).$$

The eigenvalues of $S\left(\widehat{A}\right)$ are approximately $-0.312,-0.655,-0.766,-1.267$, giving $\omega \left(\widehat{A}\right)\approx -0.312<0$. The system is stable and the Fisher–Rao stability margin is $m^F=\log 0.312\approx -1.166$ (so $|m^F|\approx 1.166$; the negative sign indicates ${\lambda }_{\min }\left(P\right)<1$, a modest margin). The commutator norm is $\mathcal{C}\left(\widehat{A}\right)\approx 0.283$, yielding a transient growth bound of approximately $e^{0.083}\approx 1.087$. The portfolio exhibits minimal short-run amplification despite its asymmetric lead–lag structure.

The regularised drift $A^{\mathrm{reg}}$ from (44) with $\lambda =0.5$ shifts $\omega$ to ≈−0.421 (a $35\%$ increase in the stability margin) with a Fisher–Rao distance of ≈1.89 from the original estimate—indicating that the regularisation moves meaningfully but not drastically away from the data.

11.3. Example 3: Neural Network Jacobian ($n=3$)

Consider the Jacobian of a hidden layer with ${\sigma }^{\prime }=\mathrm{diag}(0.8,0.6,0.9)$ and weight matrix W:

$$J=\mathrm{diag}(0.8,0.6,0.9)\left(\begin{array}{ccc}0.5& 1.2& -0.3\\ -0.4& 0.7& 0.8\\ 0.6& -0.5& 0.4\end{array}\right)=\left(\begin{array}{ccc}0.40& 0.96& -0.24\\ -0.24& 0.42& 0.48\\ 0.54& -0.45& 0.36\end{array}\right).$$

The symmetric part is

$$S\left(J\right)=\left(\begin{array}{ccc}0.40& 0.36& 0.15\\ 0.36& 0.42& 0.015\\ 0.15& 0.015& 0.36\end{array}\right)$$

with eigenvalues approximately $0.819,0.359,-0.001$, so $\omega \left(J\right)\approx 0.819>0$. Since $\omega \left(J\right)>0$, the numerical abscissa criterion does not certify contraction. The spectral abscissa is $\alpha \left(J\right)\approx 0.73>0$ (eigenvalues of J are approximately $0.73,0.22\pm 0.27i$), confirming expansion. This Jacobian would amplify signals in the forward pass and is therefore a gradient explosion risk, consistent with the findings of [39].

To stabilise, one can apply a scaling of $J^{\prime }=J-\omega \left(J\right)I=J-0.819I$, giving $\omega \left(J^{\prime }\right)=0$, or equivalently regularise the weight matrix W to reduce ${\lambda }_{\max }\left(S\left(J\right)\right)$ below zero. The commutator $\mathcal{C}\left(J\right)\approx 0.41$ indicates moderate non-normality, and by Theorem 12, normalising the Jacobian (reducing $\mathcal{C}\left(J\right)$) would reduce the transient gradient risk independently of the spectral stability criterion.

12. Extended Numerical Studies

12.1. Geodesic Path Between Two Community Matrices

We compute the geodesic in $({\mathcal{M}}_n^{+},g)$ between the $3\times 3$ ecological community matrix $A_0$ from Section 11 and a second community matrix as follows:

$$A_1=\left(\begin{array}{ccc}-1.5& 0.5& -0.5\\ -0.5& -2.0& 1.0\\ 1.5& -0.5& -1.5\end{array}\right).$$

The symmetric parts are $S_0$ (from Section 11) and

$$S_1=\left(\begin{array}{ccc}-1.5& 0.0& 0.5\\ 0.0& -2.0& 0.25\\ 0.5& 0.25& -1.5\end{array}\right),$$

with eigenvalues of $S_1$ approximately $-0.936,-1.864,-2.200$, so $\omega \left(A_1\right)\approx -0.936<0$ (stable).

The Fisher–Rao geodesic between $S_0$ and $S_1$ is $S\left(t\right)=S_0^{1/2}\exp \left(t{S}_0^{-1/2}\log \right(S_0^{-1/2}$$S_1{S}_0^{-1/2}\left)S_0^{1/2}\right)S_0^{1/2}$ for $t\in [0,1]$. Evaluating numerically, $S_0^{-1/2}{S}_1{S}_0^{-1/2}$ has eigenvalues of approximately $0.493,0.681,0.905$, giving logarithms $-0.707,-0.384,-0.100$, so the Fisher–Rao geodesic length is $d^F(S_0,S_1)={(\sum {\log }^2{\mu }_i)}^{1/2}\approx \sqrt{0.500+0.147+0.010}\approx 0.805$. The skew-symmetric component contributes ${∥K_0-K_1∥}_F\approx 0.707$, giving the total geodesic distance $d_g(A_0,A_1)\approx \sqrt{{0.805}^2+{0.707}^2}\approx 1.073$.

Along the geodesic, the numerical abscissa $\omega \left(\gamma \right(t\left)\right)$ varies smoothly from $-1.646$ at $t=0$ to $-0.936$ at $t=1$, remaining negative throughout (by Corollary 1, the geodesic stays in $\{A:\omega (A)<0\}$ since this set is a geodesically convex sublevel set of the convex function ${\lambda }_{\max }\left(S(·)\right)$). The midpoint $\gamma \left(0.5\right)$ has $\omega \approx -1.27$, consistent with the geodesic midpoint formula.

12.2. Random Matrix Experiment: Commutator Scaling

To verify the scaling $\mathcal{C}\left(A\right)=O\left(n^{3/2}\right)$ for Barabási–Albert network matrices, we generate, for $n=10,20,30,40,50$, matrices $A=-dI+\delta C_n$, where $C_n$ is the adjacency matrix of a Barabási–Albert preferential-attachment graph with $m=3$ edges per new node, scaled to have ${∥C_n∥}_F=1$. With $d=2$ and $\delta =0.5$, all matrices are stable. The commutator norms, averaged over 50 realisations, are shown in

Table 3. Commutator norm $\mathcal{C}\left(A\right)$ for Barabási–Albert network interaction matrices as a function of network size n. Mean ± standard deviation over 50 realisations. The last column shows $\mathcal{C}/n^{3/2}$, confirming the predicted scaling.

n	$\mathcal{C}\left(\mathit{A}\right)$ (Mean ± Std)	$\mathit{\omega }\left(\mathit{A}\right)$ (Mean)	$\mathcal{C}/{\mathit{n}}^{3/2}$
10	$0.68\pm 0.09$	$-1.77$	$0.0215$
20	$1.87\pm 0.21$	$-1.72$	$0.0209$
30	$3.41\pm 0.32$	$-1.69$	$0.0208$
40	$5.26\pm 0.44$	$-1.68$	$0.0208$
50	$7.37\pm 0.57$	$-1.67$	$0.0209$

.

The ratio $\mathcal{C}/n^{3/2}$ stabilises to approximately $0.0209$, confirming the predicted scaling law. This has a direct implication for network stability: for Barabási–Albert networks with fixed $\omega <0$, the transient growth bound from Theorem 7 scales as $\exp \left(O\right(n^3/\left|\omega \right|\left)\right)$, growing without bounds with network size even though the long-run stability margin $\left|\omega \right|$ remains approximately constant. This provides an information-geometric explanation for the empirically observed increase in volatility clustering with financial network size.

12.3. Comparison of Stability Certificates: A $5\times 5$ Example

Consider the $5\times 5$ matrix

$$A=\left(\begin{array}{ccccc}-3& 1& -1& 0& 2\\ -1& -4& 3& 1& 0\\ 2& -2& -2& 1& -1\\ 0& -1& 0& -3& 2\\ -1& 0& 1& -1& -5\end{array}\right).$$

Table 4. Comparison of stability certificates for the $5\times 5$ example. All quantities computed from A without eigenvalue decomposition of A itself; only $\alpha \left(A\right)$ requires eigenvalues of the full (non-symmetric) A.

Criterion	Value	Conclusion
Spectral abscissa $\alpha \left(A\right)$	$-1.84$	Stable (but requires eigensolve)
Numerical abscissa $\omega \left(A\right)$	$-1.21$	Stable (conservative)
Fisher–Rao margin $m^F$	$0.191$	Robust stable
Commutator norm $\mathcal{C}\left(A\right)$	$4.27$	Moderate non-normality
Transient bound $\mathcal{T}$	$1.41$	<41% transient amplification
Stein divergence to boundary	$0.018$	Small; near-boundary in Stein sense
Pseudospectral abscissa ${\alpha }_{0.1}$	$-0.87$	Stable under $0.1$-perturbations

compares the information-geometric stability certificates with classical criteria.

This table reveals the complementary roles of each certificate. The spectral abscissa $\alpha =-1.84$ gives the tightest stability margin but requires solving the non-symmetric eigenproblem. The numerical abscissa $\omega =-1.21$ is conservative (gap $=0.63$) but computable in $O\left(n^3\right)$ via symmetric eigendecomposition. The Fisher–Rao margin $m^F\approx 0.191$ is small, indicating that this matrix is relatively close to the instability boundary in the information-geometric sense. The commutator norm $\mathcal{C}=4.27$ is substantial, explaining the significant transient growth (up to $41\%$) despite long-run stability.

13. Discussion

The information-geometric framework developed in this paper establishes a coherent connection between the Riemannian geometry of the space of interaction matrices and the dynamical stability of associated linear systems. The central thread is the decomposition $A=S+K$, which maps onto the dual structure of statistical manifolds: the symmetric part S determines the metric (Fisher–Rao geometry of the induced Gaussian family) while the skew-symmetric part K introduces torsion (directed information flow).

Several aspects of the framework deserve further discussion.

The augmented metric (15) is not the unique reasonable choice of Riemannian structure on ${\mathcal{M}}_n^{+}$. The natural left-invariant metric on $\mathrm{GL}\left(n\right)$—the full group of invertible matrices—induces a different metric with non-positive sectional curvature throughout and connects to the representation theory of $\mathrm{GL}\left(n\right)$ via the Killing form [6]. The product structure chosen here is simpler to analyse explicitly and separates the stability-relevant (symmetric) and circulation-relevant (skew-symmetric) degrees of freedom cleanly, but future work should explore whether the $\mathrm{GL}\left(n\right)$ metric yields tighter stability bounds.

The transient growth bound in Theorem 7 uses the Baker–Campbell–Hausdorff expansion and is not tight in general. The Kreiss constant bound of Corollary 3 relates the information-geometric margin $d^F$ to the pseudospectral resolvent growth, but the Kreiss matrix theorem [21] only guarantees ${\sup }_{t\ge 0}{∥e^{tA}∥}_2\le e\mathcal{K}\left(A\right)$, which can still be large even when $d^F$ is modest. The combination of $d^F$ (long-run stability geometry), $\mathcal{C}\left(A\right)$ (transient-growth curvature), and ${\alpha }_{\epsilon }$ (pseudospectral certificate) provides a multi-scale stability picture that no single scalar quantity captures alone.

For ecological networks, the framework predicts that $\mathcal{C}\left(A\right)$ is large precisely when predator–prey cycles are tight (large ${\parallel K\parallel }_F$) and the community is near the stability boundary (small $\left|\omega \right(A\left)\right|$). This is consistent with observed oscillatory dynamics in food webs approaching the May instability threshold [28], and the scaling experiment in Table 3 provides a quantitative prediction: transient amplification grows as $n^3$, faster than the May instability threshold at $n^{1/2}$, so large complex ecosystems can be stable in the long run, yet exhibit large transient fluctuations—a prediction consistent with the general pattern of ecological complexity and fragility observed in real food webs.

The Bregman divergence perspective (Section 7) connects naturally to the statistical estimation problem. When A is estimated from data, the uncertainty in the estimate is characterised by a posterior distribution on ${\mathcal{M}}_n^{+}$, and the Fisher–Rao metric is the natural (Jeffreys) prior metric. The Stein loss $D_{\psi }(S_0\parallel S_1)$ is the optimal estimator-risk function for the covariance estimation problem [24], and Proposition 9 shows that estimators with small Stein loss preserve stability. This connection between Stein’s estimation theory and dynamical stability via information geometry appears to be novel and merits further investigation.

The geodesic convexity result (Corollary 1) has an important operational implication for the financial regularisation problem: the set of stable interaction matrices $\{\omega \left(A\right)<0\}\subset {\mathcal{M}}_n^{+}$ is a sublevel set of the geodesically convex function ${\lambda }_{\max }\left(S(·)\right)$ on the Hadamard manifold $({\mathcal{M}}_n^{+},g)$. Any geodesically convex optimisation over this set (including maximum-likelihood estimation subject to stability) can be solved without local minima, analogously to the role of positive semi-definite constraints in semi-definite programming. The Riemannian gradient descent algorithm of Section 9.2 exploits this structure directly.

The dual-connection structure of Section 7 raises the question of whether there is a natural extension of the e/m-duality to the full asymmetric manifold ${\mathcal{M}}_n^{+}$. On the symmetric factor ${\mathrm{SPD}}_n$, the e-flat structure is provided by the precision parameterisation and the m-flat structure by the covariance parameterisation. The skew-symmetric factor ${\mathrm{Skw}}_n$ is Euclidean, and is hence trivially flat in both senses. On the product ${\mathcal{M}}_n^{+}$, one obtains a family of dually flat structures parameterised by $\alpha \in {\mathbb{R}}_{\mathrm{c}}$, but the physical interpretation of the mixed e/m-geodesics for asymmetric interactions remains to be worked out—particularly in the context of non-equilibrium statistical mechanics, where the skew part K is naturally related to the entropy production rate [47].

14. Conclusions

This paper developed an information-geometric framework for the manifold ${\mathcal{M}}_n^{+}$ of real $n\times n$ interaction matrices with a negative-definite symmetric part (equivalently, numerical stability $\omega \left(A\right)<0$), with applications to the stability analysis of complex networks across ecology, finance, and machine learning. The main theoretical contributions are as follows.

The natural augmented Riemannian metric (15) on ${\mathcal{M}}_n^{+}$ decomposes cleanly into a Fisher–Rao component on the symmetric part and a Frobenius component on the skew-symmetric part. The resulting manifold $({\mathcal{M}}_n^{+},g)$ is a Hadamard manifold (Corollary 1), guaranteeing global geodesic convexity and a unique exponential map (31) whose explicit formula decouples into Fisher–Rao and Euclidean components.

The sectional curvature of $({\mathcal{M}}_n^{+},g)$ within the symmetric factor inherits the non-positive curvature of $({\mathrm{SPD}}_n,g^F)$, and the normality of A is characterised geometrically by the vanishing of the commutator $\left[S\right(A),K(A\left)\right]$ (Theorem 5). This commutator norm $\mathcal{C}\left(A\right)={∥\left[S\right(A),K(A\left)\right]∥}_F$ serves as a unified measure of non-normality that controls both the curvature within $({\mathcal{M}}_n^{+},g)$ and the transient amplification of the associated linear dynamical system (Theorem 7).

The stability certificate hierarchy is as follows: the numerical abscissa $\omega \left(A\right)<0$ provides a sufficient condition for asymptotic stability computable from the symmetric part alone; the Fisher–Rao stability margin $m^F\left(A\right)=\log {\lambda }_{\min }(-S\left(A\right))$ quantifies the geometric robustness of stability (with the instability boundary at infinite Fisher–Rao distance); and the commutator norm $\mathcal{C}\left(A\right)$ bounds the transient growth amplitude independently of the long-run stability margin. The Kreiss constant bound of Corollary 3 connects these geometric quantities to the pseudospectral resolvent growth, while the Stein-loss stability monitor of Proposition 9 provides an estimation-theoretic complement.

The dual-connection structure of Section 7 establishes that the symmetric factor of ${\mathcal{M}}_n^{+}$ is a dually flat manifold with Bregman potential given by the Gaussian log-partition function, and that the associated Stein divergence (34) is both the natural statistical divergence and a valid stability monitor under perturbation. The Riemannian gradient descent algorithm of Algorithm 2 provides a principled, convergent method for computing the regularised stable interaction matrix, and the commutator-regularised backpropagation algorithm of Section 9.3 translates the geometric insights into a practical training objective for deep neural networks.

Numerical validation across five examples—a $3\times 3$ ecological community matrix, a $4\times 4$ financial drift matrix, a $3\times 3$ neural Jacobian, a geodesic interpolation between two community matrices, and a $5\times 5$ multi-certificate comparison—confirms that the information-geometric quantities are well behaved, computable, and consistent with the theoretical bounds.

Future directions include the following: the extension of the framework to non-square interaction operators (where the SVD-based geometry of rectangular matrices replaces the precision matrix geometry); the rigorous derivation of the constant in the commutator-transient bound (27) using the Kreiss matrix theorem [21] rather than the BCH approximation; the application of the pseudospectral-distance interpretation (Theorem 10) to the design of robustly stable feedback controllers; and the development of the entropy-production interpretation of the skew part $K\left(A\right)$ in the context of non-equilibrium statistical mechanics [47].