On the reflexive solutions of the matrix equation $AXB+CYD=E$
Bull. Korean Math. Soc. 2009 Vol. 46, No. 3, 511-519
Printed May 1, 2009
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.
Keywords : anti-reflexive matrix, generalized reflection matrix, matrix equation, reflexive inverse, reflexive matrix
MSC numbers : 65F15, 65F20, 15A06, 15A24
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