On solving coupled Sylvester-conjugate transpose matrix equations over generalized reflexive matrices and anti-reflexive matrices

Abstract

A square matrix P is considered a generalized reflection matrix if being Hermitian and having its square equal to the identity matrix. Given two generalized reflection matrices P and Q, a matrix A is said to be reflexive (anti-reflexive) with respect to pair (P; Q) if A = PAQ (A = -PAQ). This manuscript introduces some iterative algorithms that utilizes the gradient method to solve coupled Sylvester-conjugate transpose matrix equations
over generalized reflexive matrices and anti-reflexive matrices. Furthermore, we will conduct an analysis of the convergence properties of these methods. Then, we provide numerical techniques to determine these solutions. To summarize, the numerical examples utilized in this study have effectively demonstrated the efficacy of the iterative methods presented.