Volume 16, Issue 3
Least-Squares Solutions of the Matrix Equation ATXA=B Over Bisymmetric Matrices and its Optimal Approximation

Y. Y. Zhang, Y. Lei & A. P. Liao

Numer. Math. J. Chinese Univ. (English Ser.)(English Ser.) 16 (2007), pp. 215-225

Published online: 2007-08

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  • Abstract
A real $n\times n$ symmetric matrix $X=(x_{ij})_{n \times n}$ is called a bisymmetric matrix if $x_{ij}=x_{n+1-j, n+1-i}$. Based on the projection theorem, the canonical correlation decomposition and the generalized singular value decomposition, a method useful for finding the least-squares solutions of the matrix equation $A^{T}XA=B$ over bisymmetric matrices is proposed. The expression of the least-squares solutions is given. Moreover, in the corresponding solution set, the optimal approximate solution to a given matrix is also derived. A numerical algorithm for finding the optimal approximate solution is also described.
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@Article{NM-16-215, author = { Y. Y. Zhang, Y. Lei and A. P. Liao}, title = {Least-Squares Solutions of the Matrix Equation ATXA=B Over Bisymmetric Matrices and its Optimal Approximation}, journal = {Numerical Mathematics, a Journal of Chinese Universities}, year = {2007}, volume = {16}, number = {3}, pages = {215--225}, abstract = { A real $n\times n$ symmetric matrix $X=(x_{ij})_{n \times n}$ is called a bisymmetric matrix if $x_{ij}=x_{n+1-j, n+1-i}$. Based on the projection theorem, the canonical correlation decomposition and the generalized singular value decomposition, a method useful for finding the least-squares solutions of the matrix equation $A^{T}XA=B$ over bisymmetric matrices is proposed. The expression of the least-squares solutions is given. Moreover, in the corresponding solution set, the optimal approximate solution to a given matrix is also derived. A numerical algorithm for finding the optimal approximate solution is also described.}, issn = {}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/nm/8055.html} }
TY - JOUR T1 - Least-Squares Solutions of the Matrix Equation ATXA=B Over Bisymmetric Matrices and its Optimal Approximation AU - Y. Y. Zhang, Y. Lei & A. P. Liao JO - Numerical Mathematics, a Journal of Chinese Universities VL - 3 SP - 215 EP - 225 PY - 2007 DA - 2007/08 SN - 16 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/nm/8055.html KW - AB - A real $n\times n$ symmetric matrix $X=(x_{ij})_{n \times n}$ is called a bisymmetric matrix if $x_{ij}=x_{n+1-j, n+1-i}$. Based on the projection theorem, the canonical correlation decomposition and the generalized singular value decomposition, a method useful for finding the least-squares solutions of the matrix equation $A^{T}XA=B$ over bisymmetric matrices is proposed. The expression of the least-squares solutions is given. Moreover, in the corresponding solution set, the optimal approximate solution to a given matrix is also derived. A numerical algorithm for finding the optimal approximate solution is also described.
Y. Y. Zhang, Y. Lei & A. P. Liao. (1970). Least-Squares Solutions of the Matrix Equation ATXA=B Over Bisymmetric Matrices and its Optimal Approximation. Numerical Mathematics, a Journal of Chinese Universities. 16 (3). 215-225. doi:
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