Mehdi Dehghan and Masoud Hajarian Amirkabir University of Technology

Abstract : A matrix $P\in{\mathbb C}^{n\times n}$ is called a generalized reflection matrix if $P^{\ast}=P$ and $P^{2}=I$. An $n\times n$ 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 $AXB+CYD=E$ 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.