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

Let [email protected]?C^m^x^m and [email protected]?C^n^x^n be nontrivial involution matrices; i.e. R=R^-^1 +/-I and S=S^-^1 +/-I. An mxn complex matrix A is said to be a (R,S)-symmetric ((R,S)-skew symmetric) matrix if RAS=A (RAS=-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 A"1XB"1=C,A"1X=D"1,XB"2=D"2, and A"1X=D"1,XB"2=D"2,A"3X=D"3,XB"4=D"4, 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.

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