Matrix equations over (R,S)(R,S)-symmetric and (R,S)(R,S)-skew symmetric matrices

Let R∈Cm×mR∈Cm×m and S∈Cn×nS∈Cn×n be nontrivial involution matrices; i.e. R=R−1≠±IR=R−1≠±I and S=S−1≠±IS=S−1≠±I. An m×nm×n complex matrix AA is said to be a (R,S)(R,S)-symmetric ((R,S)(R,S)-skew symmetric) matrix if RAS=ARAS=A (RAS=−ARAS=−A). The (R,S)(R,S)-symmetric and (R,S)(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=C,A1XB1=C,A1X=D1,XB2=D2, and A1X=D1,XB2=D2,A3X=D3,XB4=D4, over the (R,S)(R,S)-symmetric ((R,S)(R,S)-skew symmetric) matrix XX. We derive necessary and sufficient conditions for the existence of (R,S)(R,S)-symmetric ((R,S)(R,S)-skew symmetric) solutions for these matrix equations. Also we give the expressions for the (R,S)(R,S)-symmetric ((R,S)(R,S)-skew symmetric) solutions to the matrix equations.