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

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Qing-Feng Xiao
Xi-Yan Hu
Lei Zhang


In this paper, the following problems are discussed.

Problem I. Given matrices A ∈ Rn×m and D ∈ Rm×m, find X ∈ ASRnP such that AT XA = D, where ASRnP = {X ∈ ASRn×n|PX ∈ SRn×n for given P ∈ ORn×n satisfying PT = P}.

Problem II. Given a matrix ˜X̄∈ Rn×n, find X̂ ∈ SE such that

∥ X̄−X̂ ∥ =  infX∈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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