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The quaternion biconjugate gradient (QBiCG) method, as a novel variant of quaternion Lanczos-type methods for solving the non-Hermitian quaternion linear systems, does not yield a minimization property. This means that the method possesses a rather irregular convergence behavior, which leads to numerical instability. In this paper, we propose a new structure-preserving quaternion quasi-minimal residual method, based on the quaternion biconjugate orthonormalization procedure with coupled two-term recurrences, which overcomes the drawback of QBiCG. The computational cost and storage required by the proposed method are much less than the traditional QMR iterations for the real representation of quaternion linear systems. Some convergence properties of which are also established. Finally, we report the numerical results to show the robustness and effectiveness of the proposed method compared with QBiCG., Comment: 25 pages

The purpose of this paper is to extend the definition of the m-weak group inverse from a square matrix to a rectangular matrix, called the W-weighted m-weak group inverse. This new generalized inverse is also a generalization of the weak group inverse, generalized group inverse, Drazin inverse, weighted weak group inverse and W-weighted Drazin inverse. Furthermore, we discuss some properties, characterizations and representations of the W-weighted m-weak group inverse, as well as its applications in solving the matrix equation. Some of the results available in the paper not only recover a few results of the m-weak group inverse, weighted weak group, etc., but also some of them are novel for these generalized inverses., Comment: 25 pages, 0 figure

We employ the M-P inverses and ranks of quaternion matrices to establish the necessary and sufficient conditions for solving a system of the dual quaternion matrix equations $(AX, XC) = (B, D)$, along with providing an expression for its general solution. Serving as an application, we investigate the solutions to the dual quaternion matrix equations $AX = B$ and $XC=D$, including $\eta$-Hermitian solutions. Lastly, we design a numerical example to validate the main research findings of this paper.

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 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 extreme eigenvalue along with the associated eigenvector of the large dual quaternion Hermitian matrices. Furthermore, a convergence analysis of the Rayleigh quotient iteration is derived, demonstrating a local convergence rate of at least cubic, which is faster than the linear convergence rate of the power method. 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., Comment: arXiv admin note: text overlap with arXiv:2111.12211 by other authors

In this paper, we propose the global quaternion full orthogonalization (Gl-QFOM) and global quaternion generalized minimum residual (Gl-QGMRES) methods, which are built upon global orthogonal and oblique projections onto a quaternion matrix Krylov subspace, for solving quaternion linear systems with multiple right-hand sides. We first develop the global quaternion Arnoldi procedure to preserve the quaternion Hessenberg form during the iterations. We then establish the convergence analysis of the proposed methods, and show how to apply them to solve the Sylvester quaternion matrix equation. Numerical examples are provided to illustrate the effectiveness of our methods compared with the traditional Gl-FOM and Gl-GMRES iterations for the real representations of the original linear systems., Comment: 24pages, 4 figures

Sylvester-type matrix equations have applications in areas including control theory, neural networks, and image processing. In this paper, we establish the necessary and sufficient conditions for the system of Sylvester-type quaternion matrix equations to be consistent and derive an expression of its general solution (when it is solvable). As an application, we investigate the necessary and sufficient conditions for quaternion matrix equations to be consistent and derive a formula for its general solution involving $\eta$-Hermicity. As a special case, we also present the necessary and sufficient conditions for the system of two-sided Sylvester-type quaternion matrix equations to have a solution and derive a formula for its general solution (when it is solvable). Finally, we present an algorithm and an example to illustrate the main results of this paper., Comment: 31

This study establishes consistency conditions and a general solution for a coupled system that consists of five two-sided Sylvester-like tensor equations in ten quaternion variables throughout the Einstein tensor product. Certain specific cases are thus established. In a direct application, we investigate certain necessary and sufficient conditions for the existence of an $\eta$-Hermitian solution to five coupled two-sided Sylvester-like quaternion tensor equations. Finally, we present an algorithm and a numerical example to validate the main result., Comment: to update a new results

This paper investigates the necessary and sufficient algebraic conditions to a constrained system of Sylvester-type quaternion tensor equations. An explicit formula of the general solution regarding the Moore-Penrose inverses of some block given tensors is obtained. As an application of a particular case, we establish the solvability conditions and the general solution to a system of Sylvester-type quaternion tensor equations involving $\eta$-Hermitian unknowns. An algorithm with a numerical example is proposed to compute the general solution of the main system., Comment: to update a new results

We derive the solvability conditions and a formula of a general solution to a Sylvester-type matrix equation over Hamilton quaternions. As an application, we investigate the necessary and sufficient conditions for the solvability of the quaternion matrix equation, which involves $\eta$-Hermicity. We also provide an algorithm with a numerical example to illustrate the main results of this paper.

In this paper, we provide some solvability conditions in terms of ranks for the existence of a general solution to a system of $k$ Sylvester-type quaternion matrix equations with $3k+1$ variables $A_{i}X_{i}+Y_{i}B_{i}+C_{i}Z_{i}D_{i}+F_{i}Z_{i+1}G_{i}=E_{i},~i=\overline{1,k}$. As applications of this system, we present rank equalities as the necessary and sufficient conditions for the existence of a general solution to some systems of quaternion matrix equations $A_{i}X_{i}+(A_{i}X_{i})_{\phi}+C_{i}Z_{i}(C_{i})_{\phi}+F_{i}Z_{i+1}(F_{i})_{\phi}=E_{i},~i=\overline{1,k}$.

We derive necessary and sufficient conditions for the existence of the exact solution to the Sylvester-type quaternion tensor system $ \mathcal{A}_i\ast_{N}\mathcal{X}_i+ \mathcal{Y}_i\ast_{M}\mathcal{B}_i+\mathcal{C}_i\ast_{N} \mathcal{Z}_i\ast_{M}\mathcal{D}_i+\mathcal{F}_i\ast_{N} \mathcal{Z}_{i+1}\ast_{M}\mathcal{G}_i=\mathcal{E}_i, i=\overline{1,3} $ using Moore-Penrose inverse, and present an expression of the general solution to the system when it is solvable. As an application of this system, we provide the solvability conditions and general solutions for the Sylvester-type quaternion tensor system $ \mathcal{A}_i\ast_{N}\mathcal{Z}_i\ast_{M}\mathcal{B}_i+ \mathcal{C}_i\ast_{N}\mathcal{Z}_{i+1}\ast_{M}\mathcal{D}_i= \mathcal{E}_i, i=\overline{1,4}. $ This paper can also serve as extensions to some known results.

In this paper, we investigate some properties of annihilator $(b,c)$-inverses in an arbitrary ring. We demonstrate that one-sided annihilator $(b,c)$-inverses of elements in arbitrary rings may behave differently in contrast to one-sided $(b,c)$-inverses. Also, we discuss intertwining property, absorption law, reverse order law, and Cline's formula for annihilator $(b,c)$-inverses. As applications, we improve and extend some known results to $(b,c)$-inverses. In particular, we derive an equivalent condition of intertwining property for $(b,c)$-inverses in semigroups.

By a tensor we mean a multidimensional array (matrix) or hypermatrix over a number field. This article aims to set an account of the studies on the permanent functions of tensors. We formulate the definitions of 1-permanent, 2-permanent, and $k$-permanent of a tensor in terms of hyperplanes, planes and $k$-planes of the tensor; we discuss the polytopes of stochastic tensors; at end we present an extension of the generalized matrix function for tensors., Comment: 6th International Conference on Matrix Analysis and Applications (ICMAA), Da Nang, Vietnam, June 14-18, 2017

In this paper, we investigate and discuss in detail the structures of quaternion tensor SVD, quaternion tensor rank decomposition, and $\eta$-Hermitian quaternion tensor decomposition with the isomorphic group structures and Einstein product. Then we give the expression of the Moore-Penrose inverse of a quaternion tensor by using the quaternion tensor SVD. Moreover, we consider a generalized Sylvester quaternion tensor equation. We give some necessary and sufficient conditions for the existence of a solution to the generalized Sylvester quaternion tensor equation in terms of the Moore-Penrose inverses of the quaternion tensors. We also present the expression of the general solution to this tensor equation when it is solvable. As applications of this generalized Sylvester quaternion tensor equation, we derive some necessary and sufficient conditions for the existences of $\eta$-Hermitian solutions to some quaternion tensor equations. We also provide some numerical examples to illustrate our results.

Let $\mathbb{H}$ be the real quaternion algebra and $\mathbb{H}^{m\times n}$ denote the set of all $m\times n$ matrices over $\mathbb{H}$. Let $\mathbf{i},\mathbf{j},\mathbf{k}$ be the imaginary quaternion units. For $\eta\in\{\mathbf{i},\mathbf{j},\mathbf{k}\}$, a square real quaternion matrix $A$ is said to be $\eta$-Hermitian if $A^{\eta*}=A$ where $A^{\eta*}=-\eta A^{\ast}\eta$, and $A^{\ast}$ stands for the conjugate transpose of $A$. In this paper, we construct a simultaneous decomposition of four real quaternion matrices with the same row number $(A,B,C,D),$ where $A=A^{\eta*}\in \mathbb{H}^{m\times m}, B\in \mathbb{H}^{m\times p_{1}},C\in \mathbb{H}^{m\times p_{2}},D\in \mathbb{H}^{m\times p_{3}}$. As applications of this simultaneous matrix decomposition, we derive necessary and sufficient conditions for some real quaternion matrix equations involving $\eta$-Hermicity in terms of ranks of the coefficient matrices. We also present the general solutions to these real quaternion matrix equations. Moreover, we provide some numerical examples to illustrate our results.

In this paper, we derive some necessary and sufficient solvability conditions for some systems of one sided coupled Sylvester-type real quaternion matrix equations in terms of ranks and generalized inverses of matrices. We also give the expressions of the general solutions to these systems when they are solvable. Moreover, we provide some numerical examples to illustrate our results. The findings of this paper extend some known results in the literature.

In this paper, we give the complete structures of the equivalence canonical form of four matrices over an arbitrary division ring. As applications, we derive some practical necessary and sufficient conditions for the solvability to some systems of generalized Sylvester matrix equations using the ranks of their coefficient matrices. The results of this paper are new and available over the real number field, the complex number field, and the quaternion algebra.

We prove inequalities on non-integer powers of products of generalized matrices functions on the sum of positive semi-definite matrices. For example, for any real number $r \in \{1\} \cup [2, \infty)$, positive semi-definite matrices $A_i,\ B_i,\ C_i\in M_{n_i}$, $i=1,2$, and generalized matrix functions $d_\chi, d_\xi$ such as the determinant and permanent, etc., we have \begin{eqnarray*}&&\left(d_\chi(A_1+B_1+C_1)d_\xi(A_2+B_2+C_2)\right)^r \\ &&\hskip 1in + \left(d_\chi(A_1)d_\xi(A_2)\right)^r + \left(d_\chi(B_1)d_\xi(B_2)\right)^r + \left(d_\chi(C_1)d_\xi(C_2)\right)^r \\ & \ge &\left(d_\chi(A_1+B_1 )d_\xi(A_2+B_2 )\right)^r + \left(d_\chi(A_1+ C_1)d_\xi(A_2+ C_2)\right)^r + \left(d_\chi( B_1+C_1)d_\xi( B_2+C_2)\right)^r\,.\end{eqnarray*} A general scheme is introduced to prove more general inequalities involving $m$ positive semi-definite matrices for $m \ge 3$ that extend the results of other authors., Comment: 15 pages

Let $\mathbb{H}$ be the real quaternion algebra and $\mathbb{H}^{n\times m}$ denote the set of all $n\times m$ matrices over $\mathbb{H}$. In this paper, we construct a simultaneous decomposition of seven general real quaternion matrices with compatible sizes: $A\in \mathbb{H}^{m\times n}, B\in \mathbb{H}^{m\times p_{1}},C\in \mathbb{H}^{m\times p_{2}},D\in \mathbb{H}^{m\times p_{3}},E\in \mathbb{H}^{q_{1}\times n},F\in \mathbb{H}^{q_{2}\times n},G\in \mathbb{H}^{q_{3}\times n}$. As applications of the simultaneous matrix decomposition, we give solvability conditions, general solutions, as well as the range of ranks of the general solutions to the following two real quaternion matrix equations $BXE+CYF+DZG=A$ and $BX+WE+CYF+DZG=A,$ where $A,B,C,D,E,F,$ and $G$ are given real quaternion matrices.