Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation [formula omitted

Abstract

A matrix P ∈ R n × n is called a generalized reflection matrix if P T = P and P 2 = I . An n × n matrix A is said to be a reflexive (anti-reflexive) matrix with respect to the generalized reflection matrix P if A = P A P ( A = − P A P ) . 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.