| Least-Squares Solutions of the Matrix Equation A^T XA = B Over Bisymmetric Matrices and its Optimal Approximation |
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| 作者姓名： | Yanyan Zhang Yuan Lei Anping Liao |
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| 作者单位： | Yanyan Zhang (Hunan University of Arts and Science，Changde 415000，China) Yuan Lei Anping Liao (College of Mathematics and Econometrics，Hunan University，Changsha 410082，China) |
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| 摘 要： | A real n×n symmetric matrix X=(x_(ij))_(n×n)is called a bisymmetric matrix if x_(ij)=x_(n 1-j,n 1-i).Based on the projection theorem,the canonical correlation de- composition and the generalized singular value decomposition,a method useful for finding the least-squares solutions of the matrix equation A~TXA=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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| 关 键 词： | 轴对称矩阵 矩阵方程 典型相关分解 最小二乘法 最佳逼近 |
| 修稿时间： | 2005-12-122006-09-22 |
Least-Squares Solutions of the Matrix Equation ATXA = B Over Bisymmetric Matrices and its Optimal Approximation |
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| Yanyan Zhang Yuan Lei Anping Liao.Least-Squares Solutions of the Matrix Equation A^T XA = B Over Bisymmetric Matrices and its Optimal Approximation[J].Numerical Mathematics A Journal of Chinese Universities English Series,2007,16(3):215-225. |
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| Authors: | Yanyan Zhang Yuan Lei Anping Liao |
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| Abstract: | A real n × n symmetric matrix X = (xij)n×n is called a bisymmetric matrix if xij = xn 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 ATXA = 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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| Keywords: | Bisymmetric matrix canonical correlation decomposition generalized singular value decomposition least-squares solution optimal approximate solution |
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