THE REFLEXIVE AND ANTI-REFLEXIVE SOLUTIONS OF A LINEAR MATRIX EQUATION AND SYSTEMS OF MATRIX EQUATIONS

An n x n complex matrix P is said to be a generalized reflection matrix if P* = P and P-2 = I (where P* is the conjugate transpose of P). An n x n complex matrix A is said to be a reflexive (anti-reflexive) matrix with respect to the generalized reflection matrix P if A = PAP (A = -PAP). The reflexive and anti-reflexive matrices have wide applications in information theory, linear estimate theory and numerical analysis. In this paper, we will consider the matrix equations (I) A(1)XB(1) = D-1, (II) A(1)X = C-1, XB2 = C-2, and (III) A(1)X = C-1, XB2 = C-2, A(3)X = C-3, XB4 = C-4, over reflexive and anti-reflexive matrices. We first introduce several decompositions of A(1), B-1, C-1, B-2, C-2, A(3), C-3, B-4 and C-4, then by applying these decompositions, the necessary and sufficient conditions for the solvability of matrix equations (I), (II) and (III) over reflexive or anti-reflexive matrices are proposed. Also some general expressions of the solutions for solvable cases are obtained.