Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation $A_1{X}_1{B}_1+A_2{X}_2{B}_2=C$

Abstract

A matrix $P\in R^{n\times n}$ is called a generalized reflection matrix if $P^T=P$ and $P^2=I$. An $n\times n$ matrix $A$ is said to be a reflexive (anti-reflexive) matrix with respect to the generalized reflection matrix $P$ if $A=PAP(A=-PAP)$. In this paper, three iterative algorithms are proposed for solving the linear matrix equation $A_1{X}_1{B}_1+A_2{X}_2{B}_2=C$ over reflexive (anti-reflexive) matrices $X_1$ and $X_2$. When this matrix equation is consistent over reflexive (anti-reflexive) matrices, for any reflexive (anti-reflexive) initial iterative matrices, the reflexive (anti-reflexive) solutions can be obtained within finite iterative steps in the absence of roundoff errors. By the proposed iterative algorithms, the least Frobenius norm reflexive (anti-reflexive) solutions can be derived when spacial initial reflexive (anti-reflexive) matrices are chosen. Furthermore, we also obtain the optimal approximation reflexive (anti-reflexive) solutions to the given reflexive (anti-reflexive) matrices in the solution set of the matrix equation. Finally, some numerical examples are presented to support the theoretical results of this paper.