Search Results (20)

In this paper, we investigate a pair of Hermitian and anti-Hermitian solutions of a generalized Sylvester matrix equation $AXB+C\overline{X}D+EYF=G$ over commutative quaternion algebra by using a complex representation of commutative quaternion matrices, the Kronecker product, vec-operation, and Moore–Penrose-generalized inverse. We establish the necessary and sufficient conditions for the existence of solutions. Moreover, we derive explicit expressions when they are solvable. We also provide two numerical examples to illustrate the main results. Full article

This paper develops a structured analytical framework and a robust numerical methodology for the spectral stability of time-varying bilateral quaternion differential equations of the form $\dot{q}=A\left(t\right)q+qB\left(t\right)$. By systematically extending classical real matrix theory to non-commutative dynamical systems via exact isometric real representations, this study utilizes the Kronecker product of real adjoint matrices to rigorously elucidate the underlying tensor structure of the bilateral evolution operator. This tensor-based reformulation proves that the Floquet multipliers of the bilaterally coupled system can be strictly decoupled into the product of the spectra corresponding to the left and right unilateral subsystems. Second, a “Scalar-Vector Stability Separation Principle” based on logarithmic norms is proposed, demonstrating that the transient energy evolution of the system is governed exclusively by the Hermitian real parts of the coefficient matrices, remaining entirely independent of the anti-Hermitian imaginary parts (rotation terms). Furthermore, for constant-coefficient and slowly varying systems, the Riesz projection from holomorphic functional calculus is introduced to establish algebraic criteria for exponential dichotomies, thereby revealing a cubic scaling law that relates the robustness threshold to the spectral gap (${\epsilon }_0\propto {\beta }^3$). Numerically, a Quaternion Chebyshev Spectral Collocation Method (Q-CSCM) is embedded within this exact vectorization framework to ensure that the algebraic symmetries of the bilateral system are strictly preserved through the isomorphic mapping. By explicitly constructing the fully discrete Kronecker product matrix via the exact real vectorization isomorphism, discrete energy estimates are utilized to rigorously prove that the numerical scheme successfully inherits the intrinsic spectral accuracy of the Chebyshev approximation. Comprehensive numerical experiments demonstrate that, within the low-dimensional regime, this methodology exhibits substantial temporal approximation efficiency advantages and superior numerical robustness compared to an alternative Legendre spectral baseline, as well as traditional explicit and state-of-the-art implicit symplectic Runge–Kutta methods, particularly when solving stiff and critically stable problems such as nonlinear Riccati oscillators. Full article

This paper primarily investigates the solutions to the system of equations $A_{1}X{B}_1=C_1$ and $A_{2}X{B}_2=C_2$. This system generalizes the classical equation $AXB=C$, as well as the system of equations $AX=B$ and $XC=D$, and finds broad applications in control theory, signal processing, networking, optimization, and other related fields. Various methods for solving this system are introduced, including the generalized inverse method, the $\mathrm{vec}$-operator method, matrix decomposition techniques, Cramer’s rule, and iterative algorithms. Based on these approaches, the paper discusses general solutions, symmetric solutions, Hermitian solutions, and other special types of solutions over different algebraic structures, such as number fields, the real field, the complex field, the quaternion division ring, principal ideal domains, regular rings, strongly *-reducible rings, and operators on Banach spaces. In addition, matrix systems related to the system $A_{1}X{B}_1=C_1$ and $A_{2}X{B}_2=C_2$ are also explored. Full article

This survey provides a review of the theoretical research on the classic system of matrix equations $AX=C$ and $XB=D$, which has wide-ranging applications across fields such as control theory, optimization, image processing, and robotics. The paper discusses various solution methods for the system, focusing on specialized approaches, including generalized inverse methods, matrix decomposition techniques, and solutions in the forms of Hermitian, extreme rank, reflexive, and conjugate solutions. Additionally, specialized solving methods for specific algebraic structures, such as Hilbert spaces, Hilbert $C^{\ast }$-modules, and quaternions, are presented. The paper explores the existence conditions and explicit expressions for these solutions, along with examples of their application in color images. Full article

In this paper, based on the novel generalized Hamilton-real (GHR) calculus, we propose for the first time a quaternion Nesterov accelerated projected gradient algorithm for computing the dominant eigenvalue and eigenvector of quaternion Hermitian matrices. By introducing momentum terms and look-ahead updates, the algorithm achieves a faster convergence rate. We theoretically prove the convergence of the quaternion Nesterov accelerated projected gradient algorithm. Numerical experiments show that the proposed method outperforms the quaternion projected gradient ascent method and the traditional algebraic methods in terms of computational accuracy and runtime efficiency. Full article

The application of eigenvalue theory to dual quaternion Hermitian matrices holds significance in the realm of multi-agent formation control. In this paper, we study the use of Rayleigh quotient iteration (RQI) for solving the right eigenpairs of dual quaternion Hermitian matrices. Combined with dual representation, the RQI algorithm can effectively compute the eigenvalue along with the associated eigenvector of the dual quaternion Hermitian matrices. Furthermore, by utilizing the minimal residual property of the Rayleigh quotient, a convergence analysis of the Rayleigh quotient iteration is derived. Numerical examples are provided to illustrate the high accuracy and low CPU time cost of the proposed Rayleigh quotient iteration compared with the power method for solving the dual quaternion Hermitian eigenvalue problem. Full article

Dual algebra plays an important role in kinematic synthesis and dynamic analysis, but there are still few studies on dual quaternion matrix theory. This paper provides an efficient method for solving the QLY least squares problem of the dual quaternion matrix equation $AXB+CYD\approx E$, where X, Y are unknown dual quaternion matrices with special structures. First, we define a semi-tensor product of dual quaternion matrices and study its properties, which can be used to achieve the equivalent form of the dual quaternion matrix equation. Then, by using the dual representation of dual quaternion and the $\mathcal{GH}$-representation of special dual quaternion matrices, we study the expression of QLY least squares Hermitian solution of the dual quaternion matrix equation $AXB+CYD\approx E$. The algorithm is given and the numerical examples are provided to illustrate the efficiency of the method. Full article

Split quaternions have various applications in mathematics, computer graphics, robotics, physics, and so on. In this paper, two useful, real representations of a split quaternion matrix are proposed. Based on this, we derive their fundamental properties. Then, via the real representation method, we obtain the necessary and sufficient conditions for the existence of solutions to two split quaternion matrix equations. In addition, two experimental examples are provided to show their feasibility. Full article

In this paper, we investigate the necessary and sufficient conditions for solving a dual split quaternion matrix equation $AXB = C$, and present the general solution expression when the solvability conditions are met. As an application, we delve into the necessary and sufficient conditions for the existence of a Hermitian solution to this equation by using a newly defined real representation method. Furthermore, we obtain the solutions for the dual split quaternion matrix equations $AX = C$ and $XB = C$. Finally, we provide a numerical example to demonstrate the findings of this paper. Full article

This paper considers the Hermitian solutions of a new system of commutative quaternion matrix equations, where we establish both necessary and sufficient conditions for the existence of solutions. Furthermore, we derive an explicit general expression when it is solvable. In addition, we also provide the least squares Hermitian solution in cases where the system of matrix equations is not consistent. To illustrate our main findings, in this paper we present two numerical algorithms and examples. Full article

Dual quaternions have wide applications in automatic differentiation, computer graphics, mechanics, and others. Due to its application in control theory, matrix equation $AXB=C$ has been extensively studied. However, there is currently limited information on matrix equation $AXB=C$ regarding the dual quaternion algebra. In this paper, we provide the necessary and sufficient conditions for the solvability of dual quaternion matrix equation $AXB=C$, and present the expression for the general solution when it is solvable. As an application, we derive the $\varphi$-Hermitian solutions for dual quaternion matrix equation $AX{A}^{\varphi }=C$, where the $\varphi$-Hermitian extends the concepts of Hermiticity and $\eta$-Hermiticity. Lastly, we present a numerical example to verify the main research results of this paper. Full article

Let $\mathbb{H}$ denote the quaternion algebra. This paper investigates the generalized complementary covariance, which is the $\varphi$-Hermitian quaternion matrix. We give the properties of the generalized complementary covariance matrices. In addition, we explore the unitary diagonalization of the covariance and generalized complementary covariance. Moreover, we give the generalized quaternion unitary transform algorithm and test the performance by numerical simulation. Full article

The Dyson index, $\beta$, plays an essential role in random matrix theory, as it labels the so-called “three-fold way” that refers to the symmetries satisfied by ensembles under unitary transformations. As is known, its $1,$ 2, and 4 values denote the orthogonal, unitary, and symplectic classes, whose matrix elements are real, complex, and quaternion numbers, respectively. It functions, therefore, as a measure of the number of independent non-diagonal variables. On the other hand, in the case of $\beta$ ensembles, which represent the tridiagonal form of the theory, it can assume any real positive value, thus losing that function. Our purpose, however, is to show that, when the Hermitian condition of the real matrices generated with a given value of $\beta$ is removed, and, as a consequence, the number of non-diagonal independent variables doubles, non-Hermitian matrices exist that asymptotically behave as if they had been generated with a value $2\beta .$ Therefore, it is as if the $\beta$ index were, in this way, again operative. It is shown that this effect happens for the three tridiagonal ensembles, namely, the $\beta$–Hermite, the $\beta$–Laguerre, and the $\beta$–Jacobi ensembles. Full article

In this paper, we propose three real representations of a generalized Segre quaternion matrix. We establish necessary and sufficient conditions for the existence of the $\eta$-anti-Hermitian solution to a system of constrained matrix equations over the generalized Segre quaternion algebra. We also obtain the expression of the general $\eta$-anti-Hermitian solution to the system when it is solvable. Finally, we provide a numerical example to verify the main results of this paper. Full article

This paper investigates the properties of the $\varphi$-skew-Hermitian solution to the system of quaternion matrix equations involving $\varphi$-skew-Hermicity with four unknowns $A_i{X}_i{\left(A_i\right)}_{\varphi }+B_i{X}_{i+1}{\left(B_i\right)}_{\varphi }=C_i,(i=1,2,3),A_4{X}_4{\left(A_4\right)}_{\varphi }=C_4$. We present the general $\varphi$-skew-Hermitian solution to this system. Moreover, we derive the $\beta \left(\varphi \right)$-signature bounds of the $\varphi$-skew-Hermitian solution $X_1$ in terms of the coefficient matrices. We also give some necessary and sufficient conditions for the system to have $\beta$($\varphi$)-positive semidefinite, $\beta$($\varphi$)-positive definite, $\beta$($\varphi$)-negative semidefinite and $\beta$($\varphi$)-negative definite solutions. Full article