# The (R,S)-symmetric and (R,S)-skew symmetric solutions of the pair of matrix equations A1XB1 = C1 and A2XB2 = C2

Document Type: Research Paper

Authors

Abstract

Let $Rin textbf{C}^{mtimes m}$ and $Sin textbf{C}^{ntimes n}$ be nontrivial involution matrices; i.e., $R=R^{-1}neq pm~I$ and $S=S^{-1}neq pm~I$.
An $mtimes n$ complex matrix $A$ is said to be an $(R, S)$-symmetric ($(R, S)$-skew symmetric) matrix if $RAS =A$ ($RAS =-A$).
The $(R, S)$-symmetric and $(R, S)$-skew symmetric matrices have
a number of special properties and widely used in engineering and
scientific computating. Here, we introduce the necessary and
sufficient conditions for the solvability of the pair of matrix
equations $A_{1}XB_{1}=C_{1}$ and $A_{2}XB_{2}=C_{2}$, over $(R, S)$-symmetric and $(R, S)$-skew symmetric matrices, and give the
general expressions of the solutions for the solvable cases.
Finally, we give necessary and sufficient conditions for the
existence of $(R, S)$-symmetric and $(R, S)$-skew symmetric
solutions and representations of these solutions to the pair of
matrix equations in some special cases.

Keywords

### History

• Receive Date: 03 February 2009
• Revise Date: 15 March 2012
• Accept Date: 17 March 2010