Least squares (P,Q)-orthogonal symmetric solutions of the matrix equation and its optimal approximation

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Lin-lin Zhao
Guo-linag Chen
Qing-bin Liu

Abstract

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 AT XB = 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 AT XB = C to a given matrix is obtained. An explicit expression of the least square (P, Q)-orthogonal symmetric solution for the matrix equation AT XB = C with the minimum-norm is also derived. An algorithm for finding the optimal approximation solution is described and some numerical results are given.

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