Star-Convexity of the Eigenvalue Regions for Stochastic Matrices and Certain Subclasses

Abstract

Star-convexity of the eigenvalue region for the set of $n\times n$ stochastic matrices has already been proven, for $n\ge 2$, by Dmitriev and Dynkin. The star-convexity property enables full determination of the eigenvalue region by its boundary. This study offers a more straightforward proof that extends to other subclasses of the stochastic matrices. Furthermore, the proof is constructive as it includes the explicit construction of the corresponding realizing matrices. Explicit sufficient conditions for star-convexity of the eigenvalue regions of stochastic subclasses are presented. In particular, star-convexity of the eigenvalue region is proved for the $n\times n$ doubly stochastic and the $n\times n$ monotone stochastic matrices.

1. Introduction

This paper deals with the study of the eigenvalue region of a given set of matrices, i.e., the region in the complex plane consisting of all eigenvalues of all matrices in this set. Star-convexity can be useful in determining this region. A region is called star-convex with respect to the origin if each line segment connecting the origin to any point within the region fully belongs to this region. Consequently, when the region is star-convex, it is fully determined by its boundary.

For some sets of matrices, star-convexity of their eigenvalue regions has already been established. For example, as early as 1946, Dmitriev and Dynkin [1] proved, using an innovative geometric argument, the star-convexity of the eigenvalue region for the $n\times n$ stochastic matrices. Stochastic matrices are non-negative matrices whose rows sum to 1 and have widespread applications such as Markov chain theory [2]. The first aim of this paper is to provide a more straightforward proof of the already proven star-convexity of eigenvalue regions for stochastic matrices.

The investigation of eigenvalue regions of stochastic matrices dates back to a problem posed by Kolmogorov in 1938. Dmitriev and Dynkin made significant progress by determining these regions for stochastic matrices of order up to five. Building on their work, Karpelevich [3], in 1951, succeeded in generalising their approach to characterise the eigenvalue regions for stochastic matrices of arbitrary dimension, resulting in the well-known Karpelevich regions. While Karpelevich’s theorem and proof resolved the general case, the result is notably intricate and often challenging to apply in concrete settings, see, e.g., [4]. The complexity and subtlety of this topic continue to motivate contemporary research, as evidenced by a number of recent studies. These works explore alternative characterisations of eigenvalue regions [5], examine the geometric and analytical properties of their boundary curves [6,7], and address the realisation problem—constructing stochastic matrices whose eigenvalues lie precisely on the boundary of the region [8,9].

A subclass of stochastic matrices, whose eigenvalue region has already been established to be star-convex, is the set of the $n\times n$ circulant stochastic matrices, i.e., $n\times n$ stochastic matrices in which every row, except the first, is formed by shifting the elements of the previous row one position to the right in a cyclic manner [10]. These matrices occur, among others, in vibration models with cyclic symmetry [11].

For other subclasses of stochastic matrices, as for example for the $n\times n$ doubly stochastic matrices [12] (Chapter V), i.e., $n\times n$ stochastic matrices with each column summing to 1, and for the $n\times n$ monotone stochastic matrices [13], i.e., $n\times n$ stochastic matrices in which each row stochastically dominates the previous one, the star-convexity of the eigenvalue region has not yet been proved. Recent work has studied the eigenvalues of doubly stochastic and monotone stochastic matrices [14,15], but the star-convexity of their eigenvalues regions has not been proven so far. Therefore, the second aim of this paper is to extend the more straightforward proof of star-convexity of the eigenvalue regions from stochastic matrices to doubly stochastic and monotone stochastic matrices.

In summary, this study has a dual purpose. On the one hand, this paper provides a simpler proof of the star-convexity of the eigenvalue region for the $n\times n$ stochastic matrices. On the other hand, the given proof is adapted to the doubly stochastic matrices and the monotone stochastic matrices.

The paper is organized as follows. Section 2 presents definitions and important results of the already existing matrix theory that are used throughout the paper. In Section 3 it is shown that the verification of star-convexity can be reduced to studying the eigenvalues of irreducible diagonalisable matrices and the fixed-point matrix is constructed. This fixed-point matrix has useful properties for proving, in Section 4, the star-convexity of the eigenvalue region for the $n\times n$ stochastic matrices in a straightforward way. For clarification, the given proof is illustrated with an example. In addition, in Section 5 sufficient conditions for star-convexity of the eigenvalue regions for a stochastic subclass are explicitly described. Further, the proof of the star-convexity property for stochastic matrices is adapted to some stochastic subclasses, namely the doubly stochastic matrices and the monotone stochastic matrices. Finally, Section 6 presents the conclusions of this study and outlines directions for further research.

2. Key Definitions and Properties

Before starting with the proof of star-convexity of eigenvalue regions, we list the main state-of-the-art concepts related to this research. In this paper, we will focus only on non-negative matrices. For more information, the interested reader can consult [12,16].

An important classification of the non-negative matrices is the division into reducible and irreducible matrices.

Definition 1. 

An $n\times n$ matrix A, $n⩾2$, is called reducible if there exists a permutation matrix P, such that

$$P^{T}AP=\left(\begin{array}{cc}B& C\\ O& D\end{array}\right),$$

where B and D are square submatrices, and the notation $P^T$ refers to the transpose of P. Otherwise, A is called irreducible. For $n=1$, a $1\times 1$ matrix A is irreducible by definition.

The Perron–Frobenius theorem stands as a fundamental result in both non-negative matrix theory and spectral theory. Here, we highlight its findings restricted to irreducible matrices crucial to this study.

Theorem 1 

(The Perron–Frobenius theorem for irreducible matrices). Suppose A is an $n\times n$ non-negative irreducible matrix. Then, there exists an eigenvalue r, such that:

1. 

r is real and $r>0$;

2. 

r can be associated with strictly positive left and right eigenvectors of A;

3. 

$r⩾\left|\lambda \right|$ for any eigenvalue λ of A.

A further essential classification of non-negative matrices is their separation into diagonalisable and non-diagonalisable types.

Definition 2. 

An $n\times n$ matrix A is called diagonalisable if it can be written in the following form:

$$P^{-1}AP=\mathit{diag}({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n)=\left(\begin{array}{ccccc}{\lambda }_1& 0& 0& \cdots & 0\\ 0& {\lambda }_2& 0& \cdots & 0\\ 0& 0& {\lambda }_3& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& 0& 0& \cdots & {\lambda }_n\end{array}\right),$$

where the eigenvector matrix P is a nonsingular $n\times n$ matrix, whose columns are right eigenvectors of A, and the inverse $P^{-1}$ is an $n\times n$ matrix, whose rows are left eigenvectors of A. Otherwise, A is called non-diagonalisable.

Two matrices may also be transformed into diagonal form simultaneously.

Definition 3. 

The $n\times n$ matrices A and B are called simultaneously diagonalisable if there exists an $n\times n$ nonsingular matrix P, such that both $P^{-1}AP$ and $P^{-1}BP$ are diagonal matrices.

The following theorem presents a useful equivalence for simultaneous diagonalisability relevant to this study.

Theorem 2. 

Let A and B be $n\times n$ diagonalisable matrices. Then, $AB=BA$ if and only if A and B are simultaneously diagonalisable.

Definition 4. 

A set $\mathcal{A}$ is called convex if each convex combination belongs again to this set, i.e., for all $A_1,A_2\in \mathcal{A}$ holds that $t{A}_1+(1-t)A_2\in \mathcal{A}$ for every $t\in [0,1]$.

Further, the space $(M_n\left(\mathbb{R}\right),\parallel ·{\parallel }_{\infty })$ of all $n\times n$ matrices with real matrix elements, fitted with the infinity operator norm ${\parallel ·\parallel }_{\infty }$, is a complete metric space and thus a Baire space. For the scope of this research, the following lemma is useful. More details can be found in [17].

Lemma 1. 

X is a Baire space if and only if given any countable collection $\left\{U_n\right\}$ of open sets in X, each of which is dense in X, their intersection $\bigcap U_n$ is also dense in X.

Definition 5. 

A set $\mathcal{A}$ of $n\times n$ matrices, fitted with a norm $\parallel ·\parallel$, is called closed if all limit points belong to this set, i.e., the limit of each sequence ${\left(A_k\right)}_k$, with $A_k\in \mathcal{A}$ for all k, is again contained in the set $\mathcal{A}$.

An example of a convex and closed matrix set is the set ${\mathcal{S}}_n$ of the $n\times n$ stochastic matrices. Moreover, $({\mathcal{S}}_n,\parallel ·{\parallel }_{\infty }$) is a complete metric space as it is a closed subset of $(M_n\left(\mathbb{R}\right),\parallel ·{\parallel }_{\infty })$.

Definition 6. 

A non-negative $n\times n$ matrix S is (row)stochastic if each row sums to 1. Hence, a stochastic matrix $S=\left(s_{ij}\right)$ satisfies the following conditions:

1. 

$s_{ij}⩾0,\forall i,j\in \{1,\dots ,n\}$;

2. 

${\sum }_{j=1}^n{s}_{ij}=1,\forall i\in \{1,\dots ,n\}.$

To determine the eigenvalue region for ${\mathcal{S}}_n$ exactly, the following concept can be very useful.

Definition 7. 

A set Φ is called star-convex, with respect to a point $\kappa \in \Phi$, if for every $\lambda \in \Phi$ and every $\alpha \in [0,1]$ holds that $(1-\alpha )\kappa +\alpha \lambda \in \Phi$.

This study investigates the star-convexity of eigenvalue regions with respect to the origin. So, from now on, in the context of this research, the set $\Phi$ is an eigenvalue region and $\kappa$ is supposed to be the origin. The eigenvalue region for ${\mathcal{S}}_n$ contains zero, as it is an eigenvalue of, among others, a stochastic matrix with identical rows.

Consequently, if an eigenvalue region is star-convex, its boundary alone is sufficient to fully determine it. This motivates the more straightforward proof of the star-convexity of the eigenvalue region for the $n\times n$ stochastic matrices, presented in the following sections.

3. Reduction to Irreducible Diagonalisable Matrices

Let S be an arbitrary $n\times n$ stochastic matrix with spectrum the multiset $\sigma \left(S\right)=\{{\lambda }_1,\dots ,{\lambda }_n\}$ and denote ${\Theta }_n=\{\lambda \in \mathbb{C}:\lambda$ is an eigenvalue of an $n\times n$ stochastic matrix}. To prove that the eigenvalue region ${\Theta }_n$ is star-convex, it suffices to show that $t{\lambda }_i\in {\Theta }_n$ for $1\le i\le n$ and $0\le t\le 1$.

Lemma 2. 

The $n\times n$ irreducible diagonalisable stochastic matrices are dense in ${\mathcal{S}}_n$.

Proof. 

The set consisting of the $n\times n$ stochastic matrices with distinct eigenvalues is dense in ${\mathcal{S}}_n$ ([18]). The open subset of $n\times n$ diagonalisable stochastic matrices contains this dense set and is, therefore, itself dense in ${\mathcal{S}}_n$.

The set consisting of the $n\times n$ stochastic matrices with all matrix elements strictly positive is trivially dense in ${\mathcal{S}}_n$. The open set of the $n\times n$ irreducible stochastic matrices contains the previously mentioned dense set and is, therefore, itself dense in ${\mathcal{S}}_n$.

From Lemma 1, it then follows that the irreducible diagonalisable stochastic matrices are dense in ${\mathcal{S}}_n$. □

Lemma 2 allows for the study of eigenvalue properties to be reduced to the irreducible diagonalisable stochastic matrices. Let us denote this last set from now on as ${\mathcal{E}}_n$.

Theorem 3. 

If star-convexity holds for the set of eigenvalues of $n\times n$ irreducible diagonalisable stochastic matrices, then ${\Theta }_n$ is star-convex, for $n\ge 2$.

Proof. 

Let S be an arbitrary $n\times n$ stochastic matrix and let ${\mathcal{E}}_n$ be the subset of $n\times n$ irreducible diagonalisable stochastic matrices. Since, by Lemma 2, ${\mathcal{E}}_n$ is dense in the set of $n\times n$ stochastic matrices ${\mathcal{S}}_n$, there exists for $S\in {\mathcal{S}}_n$ a sequence $D_1,\dots ,D_k,\dots$ of matrices in ${\mathcal{E}}_n$, such that ${\lim }_{k\to +\infty }{D}_k=S$. Hence, for $\lambda \in \sigma \left(S\right)$ there exists a sequence ${\lambda }_1,\dots ,{\lambda }_k,\dots$, where ${\lambda }_k\in \sigma \left(D_k\right)$ for all k, such that ${\lim }_{k\to \infty }{\lambda }_k=\lambda$.

Under the condition that star-convexity holds for the eigenvalues ${\lambda }_k$ of the irreducible diagonalisable matrices $D_k$, for $0\le t\le 1$ there exist ${\tilde{D}}_1,\dots {\tilde{D}}_k,\dots$ in ${\mathcal{E}}_n$ with $t{\lambda }_k\in \sigma \left({\tilde{D}}_k\right)$ for all k. The set ${\mathcal{S}}_n$ is closed and, therefore, ${\lim }_{k\to +\infty }{\tilde{D}}_k$ is a stochastic matrix with eigenvalue ${\lim }_{k\to +\infty }t{\lambda }_k=t\lambda$. Hence, star-convexity holds in general for the eigenvalues of the matrices of ${\mathcal{S}}_n$. □

Therefore, for the rest of this study, we will concentrate solely on the eigenvalues of ${\mathcal{E}}_n$. This allows us to explore the spectral properties formulated in Theorem 1.

So, through Theorem 3, let $S\in {\mathcal{E}}_n$. By the Perron–Frobenius Theorem 1, S has a unique Perron–Frobenius eigenvalue ${\lambda }_1=1$, which is simple and has a corresponding right eigenvector $\mathbf{1}={(1,1,\dots ,1)}^T$ and left eigenvector $\pi$, which is a positive probability vector satisfying ${\pi }^{T}S={\pi }^T$ and ${\pi }^T\mathbf{1}=1$. Based on the left eigenvector $\pi$ which is a fixed-point of S, we can now introduce the following matrix $F_S$.

Construction 1. 

We construct the rank-one stochastic matrix $F_S$ as

$$F_S=\mathbf{1}{\pi }^T,$$

which has the property that every row of $F_S$ is equal to ${\pi }^T$. The matrix $F_S$ is called the fixed-point matrix of S, as its rows consist of the fixed-point $\pi$.

The matrix $F_S$ has some useful properties, listed below.

Property 1. 

For $S\in {\mathcal{E}}_n$, S and $F_S$ are simultaneously diagonalisable.

Proof. 

Since $F_S=\mathbf{1}{\pi }^T$ is a rank-one matrix, $S{F}_S=\left(S\mathbf{1}\right){\pi }^T=F_S$ and $F_{S}S=\mathbf{1}\left({\pi }^{T}S\right)=F_S$, it follows that $F_S$ is diagonalisable and commutes with S. Consequently, by Theorem 2, S and $F_S$ are simultaneously diagonalisable. □

Property 2. 

S and $F_S$ share the same right and left Perron–Frobenius eigenvectors.

Proof. 

On the one hand, S and $F_S$ both have $\mathbf{1}$ as the right Perron–Frobenius eigenvector since they are both stochastic matrices. On the other hand, the left Perron–Frobenius eigenvector for S is $\pi$, which is also the left Perron–Frobenius eigenvector for $F_S$, since ${\pi }^T{F}_S=\left({\pi }^T\mathbf{1}\right){\pi }^T={\pi }^T.$ □

4. Star-Convexity of the Eigenvalue Regions for Stochastic Matrices

Through Properties 1 and 2, we can assume that P is the eigenvector matrix whose first column is the right Perron-Frobenius eigenvector $\mathbf{1}$ of both S and $F_S$, and with the first row of $P^{-1}$ the left Perron-Frobenius eigenvector ${\pi }^T$.

We have

$$P^{-1}SP=\mathrm{diag}({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n),P^{-1}{F}_{S}P=\mathrm{diag}(1,0,\dots ,0).$$

Here, ${\lambda }_1,\dots ,{\lambda }_n$ are the eigenvalues of S, where ${\lambda }_1=1$ is the simple Perron-Frobenius eigenvalue. Now, we can make the following construction.

Construction 2. 

Define the family of matrices $C\left(t\right)$ as

$$C\left(t\right)=tS+(1-t)F_S,t\in [0,1].$$

Note that $C\left(t\right)$ is a stochastic matrix, since both S and $F_S$ are stochastic matrices, and the set of $n\times n$ stochastic matrices is convex. Moreover, $C\left(t\right)$ is diagonalisable, as it is the convex combination of two simultaneously diagonalisable matrices, and irreducible, as it is the convex combination of $S\in {\mathcal{E}}_n$ and the non-negative matrix $F_S$. Hence, we can conclude that $C\left(t\right)\in {\mathcal{E}}_n$.

Theorem 4. 

The eigenvalue region ${\Theta }_n$ is star-convex for $n\ge 2$.

Proof. 

According to Theorem 3, we can restrict ourselves to the irreducible diagonalisable stochastic matrices. From Property 1, it follows that an irreducible diagonalisable stochastic matrix $S\in {\mathcal{E}}_n$ and its fixed-point matrix $F_S$ are simultaneously diagonalisable. Therefore, we have

$$P^{-1}C\left(t\right)P=t\left(P^{-1}SP\right)+(1-t)\left(P^{-1}{F}_{S}P\right)$$

and, hence

$$P^{-1}C\left(t\right)P=\mathrm{diag}(t{\lambda }_1+(1-t)·1,t{\lambda }_2,\dots ,t{\lambda }_n).$$

Since ${\lambda }_1=1$, the first diagonal entry is always 1. Thus, the eigenvalues of $C\left(t\right)\in {\mathcal{E}}_n$ are

$${\mu }_1\left(t\right)=1,{\mu }_i\left(t\right)=t{\lambda }_i,i=2,\dots ,n.$$

As t varies from 0 to 1, ${\mu }_i\left(t\right)$ are the points of the line segment from 0 to ${\lambda }_i$. Hence, for each eigenvalue ${\lambda }_i$ of S, there exists a stochastic matrix with eigenvalue $t{\lambda }_i$ and the set of eigenvalues forms a star-convex set with respect to the origin.

Note that we have already proven that $t{\lambda }_i\in {\Theta }_n$ for $2\le i\le n$. All that is left is to show that $t{\lambda }_1=t\in {\Theta }_n$. This can be seen by using the identity matrix I in a construction similar to the one above:

$$\tilde{C}\left(t\right)=tS+(1-t)I,t\in ]0,1],$$

(1)

which delivers for $\tilde{C}\left(t\right)$ the eigenvalues

$${\tilde{\mu }}_1\left(t\right)=1,{\tilde{\mu }}_i\left(t\right)=t,i=2,\dots ,n.$$

Note that $\tilde{C}\left(t\right)\in {\mathcal{E}}_n$, because $\tilde{C}\left(t\right)$ is diagonalisable, as the convex combination of S and I that are simultaneously diagonalisable, and irreducible, as the convex combination of $S\in {\mathcal{E}}_n$ and the non-negative identity matrix I.   □

We emphasize that the preceding argument constitutes a constructive proof of star-convexity. Specifically, we not only demonstrate that $t{\lambda }_i\in {\Theta }_n$ for all $1\le i\le n$ and $0\le t\le 1$, but also explicitly construct a realizing stochastic matrix for each $t{\lambda }_i$, which means a stochastic matrix having $t{\lambda }_i$ as an eigenvalue.

To give more insight in the presented proof of star-convexity of the eigenvalue region for the $n\times n$ stochastic matrices, we elaborate the reasoning for the $4\times 4$ diagonalisable, irreducible stochastic matrix $S\in {\mathcal{E}}_4$. For simplicity, we use approximated (rounded) values in the vector and matrix calculations.

$$S=\left[\begin{array}{cccc}0.31& 0.11& 0.33& 0.25\\ 0.29& 0.33& 0.36& 0.02\\ 0.18& 0.35& 0.40& 0.07\\ 0.43& 0.41& 0.07& 0.09\end{array}\right]$$

The left Perron–Frobenius probability eigenvector of S is given by:

$${\pi }^T=\left[\begin{array}{cccc}0.27& 0.29& 0.33& 0.11\end{array}\right].$$

So, the fixed-point matrix $F_S$ is:

$$F_S=\left[\begin{array}{cccc}0.27& 0.29& 0.33& 0.11\\ 0.27& 0.29& 0.33& 0.11\\ 0.27& 0.29& 0.33& 0.11\\ 0.27& 0.29& 0.33& 0.11\end{array}\right].$$

For S and $F_S$, the diagonal forms are given by:

$$D_S=P^{-1}SP=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0.2705& 0& 0\\ 0& 0& -0.0703+0.1307i& 0\\ 0& 0& 0& -0.0703-0.1307i\end{array}\right]\mathrm{and}$$

$$D_{F_S}=P^{-1}{F}_{S}P=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right].$$

Consequently, the matrix $C\left(t\right)=tS+(1-t)F_S,t\in [0,1]$ has the following diagonal form:

$$\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0.2705·t& 0& 0\\ 0& 0& (-0.0703+0.1307i)·t& 0\\ 0& 0& 0& (-0.0703-0.1307i)·t\end{array}\right]$$

Thus, the matrix $C\left(t\right)$ has the eigenvalues ${\mu }_1\left(t\right)=1,{\mu }_2\left(t\right)=0.2705·t,$ ${\mu }_3\left(t\right)=(-0.0703+0.1307i)·t$ and ${\mu }_4\left(t\right)=(-0.0703-0.1307i)·t$.

In Figure 1, the eigenvalues of S are represented by red points, and the eigenvalues of the stochastic family $C\left(t\right)=tS+(1-t)F_S$ are represented by blue points. The plot shows how the eigenvalues ${\lambda }_2,{\lambda }_3,{\lambda }_4$ of $tS+(1-t)F_S$ evolve from the eigenvalues of S to the eigenvalues of $F_S$, thereby illustrating the star-convexity of the eigenvalues for the matrix S.

5. Star-Convexity of the Eigenvalue Regions for Doubly Stochastic Matrices and Monotone Stochastic Matrices

5.1. Sufficient Conditions for Star-Convexity

For any class $\mathcal{P}$ of stochastic matrices, the eigenvalue region is a subset of the star-convex region ${\Theta }_n$. However, a subset of a star-convex set is not necessarily star-convex. So, to gain insight into whether the proof of the star-convexity of the eigenvalue regions for stochastic matrices can be adapted to stochastic subclasses, we review the relevant requirements from Section 3, Section 4 and Section 5. We discuss the five requirements for a stochastic subclass $\mathcal{P}$ one by one below.

• $\mathcal{P}$ is closed. To reduce the reasoning to irreducible diagonalisable stochastic matrices, it is required in Theorem 3 that the limit of a row of stochastic matrices is again stochastic. A stochastic subset $\mathcal{P}$ for which the limit of each row of matrices of $\mathcal{P}$ belongs to $\mathcal{P}$ is thus eligible to prove star-convexity of its eigenvalue region.
• Density of the irreducible diagonalisable matrices of $\mathcal{P}$ in $\mathcal{P}$. To make the reduction to the irreducible diagonalisable stochastic matrices, we need this density argument in Theorem 3. Thus, for $\mathcal{P}$, this density property must also hold in order to modify the given proof.
• The fixed-point matrix $F_S$ belongs to $\mathcal{P}$. For Construction 2, it is required that the fixed-point matrix $F_S$ is contained in $\mathcal{P}$.
• Convex matrix set. The convexity of $\mathcal{P}$ ensures that the matrix $C\left(t\right)=tS+(1-t)F_S,t\in [0,1]$, see Construction 2, which gives us a matrix with the desired eigenvalues, is an element of $\mathcal{P}$.
• The identity matrix I is contained in $\mathcal{P}$. To prove that the line segment $[0,1]$ also belongs to the eigenvalue region of stochastic matrices, the matrices $\tilde{C}\left(t\right)$ are introduced in (1) in Theorem 4 and use that the identity matrix I is stochastic. Whenever $\mathcal{P}$ contains I, the line segment $[0,1]$ lies within its eigenvalue region.

5.2. Star-Convexity of the Eigenvalue Regions for the Set of Doubly Stochastic Matrices

An $n\times n$ stochastic matrix T is doubly stochastic if each column sums to 1, i.e., ${\sum }_{i=1}^n{t}_{ij}=1,\forall j\in \{1,\dots ,n\}$. For each n, we denote ${\Delta }_n=\{\lambda \in \mathbb{C}:\lambda$ is an eigenvalue of an $n\times n$ doubly stochastic matrix}. From the definition, it follows that the identity matrix I is a doubly stochastic matrix. Furthermore, in an analogous way to Lemma 2, the irreducible diagonalisable doubly stochastic matrices are dense in the whole set of doubly stochastic matrices. Additionally, this set of matrices is convex and closed. Moreover, the left Perron–Frobenius probability eigenvector is ${\pi }^T=(1/n,1/n,\dots ,1/n)$, which makes the fixed-point matrix $F_S$ equal to

$$F_S=\left(\begin{array}{ccc}1/n& \cdots & 1/n\\ ⋮& \ddots & ⋮\\ 1/n& \cdots & 1/n\end{array}\right),$$

which is again a doubly stochastic matrix. Therefore, the sufficient conditions formulated in Section 5.1 are satisfied, and Theorem 5 follows.

Theorem 5. 

The eigenvalue region ${\Delta }_n$ is star-convex for $n\ge 2$.

5.3. Star-Convexity of the Eigenvalue Regions for the Set of Monotone Stochastic Matrices

A monotone stochastic matrix $M=\left(m_{ij}\right)$ is a stochastic matrix satisfying the following conditions: ${\sum }_{j=r}n{m}_{lj}⩾{\sum }_{j=r}n{m}_{kj}\forall l>k,\forall r\in \{1,\dots ,n\}$ [13]. For each n, we denote ${\Xi }_n=\{\lambda \in \mathbb{C}:\lambda$ is an eigenvalue of an $n\times n$ monotone matrix}. From the definition, it follows that the identity matrix I is a monotone stochastic matrix. Furthermore, analogous to Lemma 2, the irreducible diagonalisable monotone stochastic matrices are dense in the whole set of monotone stochastic matrices. Additionally, this set of matrices is convex and closed. Moreover, the fixed-point matrix $F_S$, with all rows equal to the left Perron–Frobenius vector ${\pi }^T$, is obviously also monotone. So, the following theorem is proven.

Theorem 6. 

The eigenvalue region ${\Xi }_n$ is star-convex for $n\ge 2$.

6. Conclusions and Further Research

This research proves star-convexity of the eigenvalue region for the $n\times n$ stochastic matrices. This paper serves multiple purposes.

Firstly, this study provides an insightful proof for the $n\times n$ stochastic matrices, for which this star-convexity property is already known from [1]. Moreover, the proof is constructive as it includes the explicit construction of the corresponding realizing matrices.

Additionally, in Section 5.1, this study provides easily verifiable sufficient conditions for star-convexity of the eigenvalue regions for stochastic subclasses.

Finally, the star-convexity property is proved for some stochastic subclasses, namely the $n\times n$ doubly stochastic matrices and the $n\times n$ monotone stochastic matrices.

The proof of the star-convexity of the eigenvalue regions of stochastic matrices was previously established by Dmitriev and Dynkin [1], who employed a geometric argument based on the fact that a stochastic matrix has nonnegative elements and row sums equal to one. Their proof relies purely on geometric reasoning to demonstrate star-convexity and is not based on the values of the matrix elements themselves. Hence, the proof as such is not applicable for stochastic subclasses characterised by extra restrictions on the matrix elements. In contrast, our method provides an alternative proof that builds the eigenvalue regions constructively, allowing the property of star-convexity to emerge more organically, as illustrated in Figure 1. Furthermore, unlike the approach of Dmitriev and Dynkin, our method extends naturally to stochastic subclasses, as it directly incorporates matrices from the subclasses under study in Construction 2. As a result, it preserves more structural information.

In future research, the listed sufficient conditions in Section 5.1 can be used to determine whether for other matrix sets of interest the eigenvalue regions are star-convex. Additionally, determining the eigenvalue regions for certain stochastic subclasses remains an open problem. The property of star-convexity allows one to focus solely on determining the boundaries of these regions.