An efficient iterative algorithm is presented to solve a system of linear matrix equations $A_{\mathrm{1}}{X}_{\mathrm{1}}{B}_{\mathrm{1}}+A_{\mathrm{2}}{X}_{\mathrm{2}}{B}_{\mathrm{2}}=E$, $C_{\mathrm{1}}{X}_{\mathrm{1}}{D}_{\mathrm{1}}+C_{\mathrm{2}}{X}_{\mathrm{2}}{D}_{\mathrm{2}}=F$ with real matrices $X_{\mathrm{1}}$ and $X_{\mathrm{2}}$. By this iterative algorithm, the solvability of the system can be determined automatically. When the system is consistent, for any initial matrices $X_{\mathrm{1}}^{\mathrm{0}}$ and $X_{\mathrm{2}}^{\mathrm{0}}$, a solution can be obtained in the absence of roundoff errors, and the least norm solution can be obtained by choosing a special kind of initial matrix. In addition, the unique optimal approximation solutions ${\widehat{X}}_{\mathrm{1}}$ and ${\widehat{X}}_{\mathrm{2}}$ to the given matrices ${\tilde{X}}_{\mathrm{1}}$ and ${\tilde{X}}_{\mathrm{2}}$ in Frobenius norm can be obtained by finding the least norm solution of a new pair of matrix equations $A_{\mathrm{1}}{\overline{X}}_{\mathrm{1}}{B}_{\mathrm{1}}+A_{\mathrm{2}}{\overline{X}}_{\mathrm{2}}{B}_{\mathrm{2}}=\overline{E},C_{\mathrm{1}}{\overline{X}}_{\mathrm{1}}{D}_{\mathrm{1}}+C_{\mathrm{2}}{\overline{X}}_{\mathrm{2}}{D}_{\mathrm{2}}=\overline{F}$, where $\overline{E}=E-A_{\mathrm{1}}{\tilde{X}}_{\mathrm{1}}{B}_{\mathrm{1}}-A_{\mathrm{2}}{\tilde{X}}_{\mathrm{2}}{B}_{\mathrm{2}}$, $\overline{F}=F-C_{\mathrm{1}}{\tilde{X}}_{\mathrm{1}}{D}_{\mathrm{1}}-C_{\mathrm{2}}{\tilde{X}}_{\mathrm{2}}{D}_{\mathrm{2}}$. The given numerical example demonstrates that the iterative algorithm is efficient. Especially, when the numbers of the parameter matrices $A_{\mathrm{1}},A_{\mathrm{2}},B_{\mathrm{1}},B_{\mathrm{2}},C_{\mathrm{1}},C_{\mathrm{2}},D_{\mathrm{1}},D_{\mathrm{2}}$ are large, our algorithm is efficient as well.

We consider the Hermitian $R$-conjugate generalized Procrustes problem to find Hermitian $R$-conjugate matrix $X$ such that ${\sum }_{k=1}^p{\parallel A_{k}X-C_k\parallel }^2\hspace{0.17em}+\hspace{0.17em}{\sum }_{l=1}^q{\parallel X{B}_l-D_l\parallel }^2$ is minimum, where $A_k$, $C_k$, $B_l$, and $D_l$ ($k=\mathrm{1,2},\dots ,p$, $l=\mathrm{1},\dots ,q$) are given complex matrices, and $p$ and $q$ are positive integers. The expression of the solution to Hermitian $R$-conjugate generalized Procrustes problem is derived. And the optimal approximation solution in the solution set for Hermitian $R$-conjugate generalized Procrustes problem to a given matrix is also obtained. Furthermore, we establish necessary and sufficient conditions for the existence and the formula for Hermitian $R$-conjugate solution to the linear system of complex matrix equations $A_{\mathrm{1}}X=C_{\mathrm{1}}$, $A_{\mathrm{2}}X=C_{\mathrm{2}},\dots ,A_{p}X=C_p$, $X{B}_{\mathrm{1}}=D_{\mathrm{1}},\dots ,X{B}_q=D_q$ ($p$ and $q$ are positive integers). The representation of the corresponding optimal approximation problem is presented. Finally, an algorithm for solving two problems above is proposed, and the numerical examples show its feasibility.

We consider the nonlinear matrix equation $X=Q+A^{\ast }{(\widehat{X}-C)}^{-1}A$, where $Q$ is positive definite, $C$ is positive semidefinite, and $\widehat{X}$ is the block diagonal matrix defined by $\widehat{X}=\text{d}\text{i}\text{a}\text{g}(X,X,\dots ,X)$. We prove that the equation has a unique positive definite solution via variable replacement and fixed point theorem. The basic fixed point iteration for the equation is given.

This note is concerned with the linear matrix equation $X=A{X}^{\top }\mathrm{B\hspace{0.17em}}+\mathrm{\hspace{0.17em}C}$, where the operator $(·{)}^{\top }$ denotes the transpose ($\top$) of a matrix. The first part of this paper sets forth the necessary and sufficient conditions for the unique solvability of the solution $X$. The second part of this paper aims to provide a comprehensive treatment of the relationship between the theory of the generalized eigenvalue problem and the theory of the linear matrix equation. The final part of this paper starts with a brief review of numerical methods for solving the linear matrix equation. In relation to the computed methods, knowledge of the residual is discussed. An expression related to the backward error of an approximate solution is obtained; it shows that a small backward error implies a small residual. Just like the discussion of linear matrix equations, perturbation bounds for solving the linear matrix equation are also proposed in this work.

An iterative method to compute the least-squares solutions of the matrix $AXB=C$ over the norm inequality constraint is proposed. For this method, without the error of calculation, a desired solution can be obtained with finitely iterative step. Numerical experiments are performed to illustrate the efficiency and real application of the algorithm.

Consider the nonlinear matrix equation $X-{\sum }_{i=\mathrm{1}}^m\mathrm{‍}{A}_i^{\mathrm{*}}{X}^{p_i}{A}_i=Q$ with . Two perturbation bounds and the backward error of an approximate solution to the equation are derived. Explicit expressions of the condition number for the equation are obtained. The theoretical results are illustrated by numerical examples.

We consider the low-rank approximation problem arising in the generalized Karhunen-Loeve transform. A sufficient condition for the existence of a solution is derived, and the analytical expression of the solution is given. A numerical algorithm is proposed to compute the solution. The new algorithm is illustrated by numerical experiments.