THE (R,S)-SYMMETRIC AND (R,S)-SKEW SYMMETRIC SOLUTIONS OF THE PAIR OF MATRIX EQUATIONS A1XB1 = C1 AND A2XB2 = C2

Let R 2 C m m and S 2 C n n be nontrivial involu- tion matrices; i.e., R = R 1 6 I and S = S 1 6 I. An m 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 scientic com- putating. Here, we introduce the necessary and sucient conditions for the solvability of the pair of matrix equations A1XB1 = C1 and A2XB2 = C2, 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 sucient conditions for the existence of (R;S)-symmetric and (R;S)-skew symmetric solutions and representations of these solutions to the pair of ma- trix equations in some special cases.

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