| 矩阵方程ATXB+BTXTA=D的极小范数最小二乘解 |
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| 引用本文： | 袁永新,戴华. 矩阵方程ATXB+BTXTA=D的极小范数最小二乘解[J]. 高等学校计算数学学报, 2005, 27(3): 232-238 |
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| 作者姓名： | 袁永新 戴华 |
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| 作者单位： | 南京航空航天大学理学院,南京,210016/华东船舶工业学院数理系,镇江,212003;南京航空航天大学理学院,南京,210016 |
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| 摘 要： | 1引言本文用Rm×n表示所有m×n实矩阵全体,ORn×n,ASRn×n分别表示n×n实正交矩阵类与反对称矩阵类.‖·‖F表示矩阵的Frobenius范数,A+为矩阵A的Moore-Penrose广义逆,A*B与A(?)B分别表示矩阵4与B的Hadamard乘积及Kronecker乘积,即若A=(aij),B=(bij),则A*B=(ajibij),A(?)B=(aijB),vec4表示矩阵A的按行拉直,即若A=[aT1,aT2,…,aTm],其中ai为A的行向量,则vecA=(a1a2…am)T.设A∈Rn×m,B∈Rp×m,D∈Rm×m,我们考虑不相容线性矩阵方程ATXB+BTXTA=D(1.1)
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| 关 键 词： | 极小范数最小二乘解 Moore-Penrose广义逆 矩阵方程 Frobenius范数 表示矩阵 矩阵类 实矩阵 反对称 矩阵A |
| 收稿时间： | 2003-11-21 |
| 修稿时间： | 2003-11-21 |
THE LEAST SQUARES SOLUTION WITH THE MINIMUM NORM OF MATRIX EQUATION ATXB + BTXTA = D |
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| Yuan Yongxin,Dai Hua. THE LEAST SQUARES SOLUTION WITH THE MINIMUM NORM OF MATRIX EQUATION ATXB + BTXTA = D[J]. Numerical Mathematics A Journal of Chinese Universities, 2005, 27(3): 232-238 |
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| Authors: | Yuan Yongxin Dai Hua |
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| Abstract: | Starting with the normal equation of inconsistent linear matrix equation ATXB + BTXTA = D, we obtain the least squares solution (X, Y) with the minimum norm of the equation. It relies on the SVD and generalized SVD of the coefficient matrices and has complexity proportional to the cost of these SVD8. |
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| Keywords: | matrix equation least squares solution minimum norm solution singular-value decomposition (SVD) generalized singular-value decomposition (GSVD). |
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