The anti-symmetric ortho-symmetric solutions of the matrix equation A^TXA=D

Abstract

Abstract. In this paper, the following problems are discussed. Problem I. Given matrices A ∈ R n×m and D ∈ R m×m , find X ∈ ASR n P such that A T XA = D, whereProblem II. Given a matrixX ∈ R n×n , findX ∈ S E such thatwhere · is the Frobenius norm, and S E is the solution set of Problem I. Expressions for the general solution of Problem I are derived. Necessary and sufficient conditions for the solvability of Problem I are provided. For Problem II, an expression for the solution is given as well.

The Generalized Bisymmetric (Bi-Skew-Symmetric) Solutions of a Class of Matrix Equations and Its Least Squares Problem

“…For Lemma 8, Wang and Yu [11] just gave the conclusion; we prove it here. The proof of Lemma 9 can be seen in [12]; for the convenience of the reader, we rewrite it.…”