Geodesic
Least squares (P,Q)-orthogonal symmetric solutions of the matrix equation and its optimal approximation
The electronic journal of linear algebra, Tome 20 (2010), pp. 537-551.

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Summary: In this paper, the relationship between the (P, Q)-orthogonal symmetric and symmetric matrices is derived. By applying the generalized singular value decomposition, the general expression of the least square (P, Q)-orthogonal symmetric solutions for the matrix equation A TXB = C is provided. Based on the projection theorem in inner space, and by using the canonical correlation decomposition, an analytical expression of the optimal approximate solution in the least squares (P, Q)-orthogonal symmetric solution set of the matrix equation A TXB = C to a given matrix is obtained. An explicit expression of the least square (P, Q)-orthogonal symmetric solution for the matrix equation A TXB = C with the minimum-norm is also derived. An algorithm for finding the optimal approximation solution is described and some numerical results are given.
Classification : 65F15, 65F20
Keywords: matrix equation, least squares solution, (P, Q)-orthogonal symmetric matrix, optimal approximate solution 
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     author = {Zhao, Lin-Lin and Chen, Guo-Linag and Liu, Qing-Bin},
     title = {Least squares {(P,Q)-orthogonal} symmetric solutions of the matrix equation and its optimal approximation},
     journal = {The electronic journal of linear algebra},
     pages = {537--551},
     publisher = {mathdoc},
     volume = {20},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ELA_2010__20__a15/}
}
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Zhao, Lin-Lin; Chen, Guo-Linag; Liu, Qing-Bin. Least squares (P,Q)-orthogonal symmetric solutions of the matrix equation and its optimal approximation. The electronic journal of linear algebra, Tome 20 (2010), pp. 537-551. http://geodesic.mathdoc.fr/item/ELA_2010__20__a15/