Geodesic
The anti-symmetric ortho-symmetric solutions of the matrix equation $A^{T}$XA=D
The electronic journal of linear algebra, Tome 18 (2009), pp. 21-29.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: In this paper, the following problems are discussed. Problem I. Given matrices A $\in R$ n$\times m$ and D $\in R$ m$\times m$ , find X $\in $ASR n Psuch that A TXA = D, where ASR n P= X $\in $ASR n$\times n$ |P X $\in $SR n$\times n$ for given P $\in $OR n$\times n$ satisfying P T= P . Problem II. Given a matrix X $\in R$ n$\times n$ , find $\hat X \in S$ Esuch that X - $\hat X = inf$ X $\in S$ EX - X , where $\cdot $is the Frobenius norm, and S Eis 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.
Classification : 65F15, 65F20
Keywords: anti-symmetric ortho-symmetric matrix, matrix equation, matrix nearness problem, optimal approximation, least-square solutions 
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     author = {Xiao, Qing-Feng and Hu, Xi-Yan and Zhang, Lei},
     title = {The anti-symmetric ortho-symmetric solutions of the matrix equation $A^{T}${XA=D}},
     journal = {The electronic journal of linear algebra},
     pages = {21--29},
     publisher = {mathdoc},
     volume = {18},
     year = {2009},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ELA_2009__18__a55/}
}
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Xiao, Qing-Feng; Hu, Xi-Yan; Zhang, Lei. The anti-symmetric ortho-symmetric solutions of the matrix equation $A^{T}$XA=D. The electronic journal of linear algebra, Tome 18 (2009), pp. 21-29. http://geodesic.mathdoc.fr/item/ELA_2009__18__a55/