ON THE REFLEXIVE SOLUTIONS OF THE MATRIX EQUATION AXB + CYD = E

Abstract

A matrix <TEX>$P{\in}\mathbb{C}^{n{\times}n}$</TEX> is called a generalized reflection matrix if <TEX>$P^*$</TEX> = P and <TEX>$P^2$</TEX> = I. An <TEX>$n{\times}n$</TEX> complex matrix A is said to be a reflexive (anti-reflexive) matrix with respect to the generalized reflection matrix P if A = PAP (A = -PAP). It is well-known that the reflexive and anti-reflexive matrices with respect to the generalized reflection matrix P have many special properties and widely used in engineering and scientific computations. In this paper, we give new necessary and sufficient conditions for the existence of the reflexive (anti-reflexive) solutions to the linear matrix equation AXB + CY D = E and derive representation of the general reflexive (anti-reflexive) solutions to this matrix equation. By using the obtained results, we investigate the reflexive (anti-reflexive) solutions of some special cases of this matrix equation.