Matrix equations over .R; S/-symmetric and .R; S/-skew symmetric matrices

Let R 2 Cm m and S 2 Cn n be nontrivial involution matrices; i.e. R D R􀀀1 6D I and S D S􀀀1 6D I. An m n complex matrix A is said to be a .R; S/-symmetric (.R; S/- skew symmetric) matrix if RAS D A (RAS D 􀀀A). The .R; S/-symmetric and .R; S/-skew symmetric matrices have many special properties and are widely used in engineering and scientific computations. In this paper, we consider the matrix equations A1XB1 D C; A1X D D1; XB2 D D2; and A1X D D1; XB2 D D2; A3X D D3; XB4 D D4; over the .R; S/-symmetric (.R; S/-skew symmetric) matrix X. We derive necessary and sufficient conditions for the existence of .R; S/-symmetric (.R; S/-skew symmetric) solutions for these matrix equations. Also we give the expressions for the .R; S/-symmetric (.R; S/-skew symmetric) solutions to the matrix equations.