The { P , Q , k + 1 } -Reflexive Solution to System of Matrix Equations A X = C , X B = D

Abstract EN

Let P ∈ C m × m and Q ∈ C n × n be Hermitian and { k + 1 } -potent matrices; that is, P k + 1 = P = P ⁎ and Q k + 1 = Q = Q ⁎ , where · ⁎ stands for the conjugate transpose of a matrix.

A matrix X ∈ C m × n is called { P , Q , k + 1 } -reflexive (antireflexive) if P X Q = X ( P X Q = - X ) .

In this paper, the system of matrix equations A X = C and X B = D subject to { P , Q , k + 1 } -reflexive and antireflexive constraints is studied by converting into two simpler cases: k = 1 and k = 2 .

We give the solvability conditions and the general solution to this system; in addition, the least squares solution is derived; finally, the associated optimal approximation problem for a given matrix is considered.