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

## Main Article Content

## 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, where ASR^{n}_{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.

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