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

Abstract

In this paper, the following problems are discussed.

Problem I. Given matrices A ∈ R^n×m and D ∈ R^m×m, find X ∈ ASRⁿ_P such that A^T XA = D, where ASRⁿ_P = {X ∈ ASR^n×n|PX ∈ SR^n×n for given P ∈ OR^n×n satisfying P^T = P}.

Problem II. Given a matrix ˜X̄∈ R^n×n, find X̂ ∈ S_E such that

∥ X̄−X̂ ∥ =  inf_X∈SE  ∥ X̄−X̂ ∥,

where ∥·∥ is the Frobenius norm, and SE 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.