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.

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**

Volume 37, No. 3

September and October 2011

Pages 269-279

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