Dehghan, M., Hajarian, M. (2011). The (R,S)-symmetric and (R,S)-skew symmetric solutions of the pair of matrix equations A1XB1 = C1 and A2XB2 = C2. Bulletin of the Iranian Mathematical Society, 37(No. 3), 269-279.

M. Dehghan; M. Hajarian. "The (R,S)-symmetric and (R,S)-skew symmetric solutions of the pair of matrix equations A1XB1 = C1 and A2XB2 = C2". Bulletin of the Iranian Mathematical Society, 37, No. 3, 2011, 269-279.

Dehghan, M., Hajarian, M. (2011). 'The (R,S)-symmetric and (R,S)-skew symmetric solutions of the pair of matrix equations A1XB1 = C1 and A2XB2 = C2', Bulletin of the Iranian Mathematical Society, 37(No. 3), pp. 269-279.

Dehghan, M., Hajarian, M. The (R,S)-symmetric and (R,S)-skew symmetric solutions of the pair of matrix equations A1XB1 = C1 and A2XB2 = C2. Bulletin of the Iranian Mathematical Society, 2011; 37(No. 3): 269-279.

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

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

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.