| The least squares problem of the matrix equation A1X1B1^T ＋ A2X2B2^T = T |
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| 作者单位： | QIU Yu-yang(Department of Mathematics, Zhejiang University, Yuquan Campus, Hangzhou 310027, China;College of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China);ZHANG Zhen-yue(Department of Mathematics, Zhejiang University, Yuquan Campus, Hangzhou 310027, China);WANG An-ding(College of Information and Electronics, Zhejiang Gongshang University, Hangzhou 310018, China) |
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| 基金项目： | The work of the first author was supported in part by the Social Science Foundation of Ministry of Education，the National Science Foundation for Young Scholars，the Natural Science Foundation of Zhejiang Province，Foundation of Education Department of Zhejiang Province，the Young Talent Foundation of Zhejiang Gongshang University |
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| 摘 要： | The matrix least squares （LS） problem minx ｜｜AXB^T--T｜｜F is trivial and its solution can be simply formulated in terms of the generalized inverse of A and B. Its generalized problem minx1,x2 ｜｜A1X1B1^T ＋ A2X2B2^T - T｜｜F can also be regarded as the constrained LS problem minx=diag（x1,x2） ｜｜AXB^T -T｜｜F with A = A1, A2] and B = B1, B2]. The authors transform T to T such that min x1,x2 ｜｜A1X1B1^T＋A2X2B2^T -T｜｜F is equivalent to min x=diag（x1 ,x2） ｜｜AXB^T - T｜｜F whose solutions are included in the solution set of unconstrained problem minx ｜｜AXB^T - T｜｜F. So the general solutions of min x1,x2 ｜｜A1X1B^T ＋ A2X2B2^T -T｜｜F are reconstructed by selecting the parameter matrix in that of minx ｜｜AXB^T - T｜｜F.
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| 关 键 词： | 矩阵方程 最小二乘问题 最小二乘法 无约束问题 参数矩阵 广义逆 母猪 变换t |
The least squares problem of the matrix equation A1X1B1T + A2X2B2T = T |
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| Authors: | Yu-yang Qiu Zhen-yue Zhang and An-ding Wang |
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| Institution: | [1]Department of Mathematics, Zhejiang University, Yuquan Campus, Hangzhou 310027, China [2]College of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310018, China [3]College of Information and Electronics, Zhejiang Gongshang University, Hangzhou 310018, China |
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| Abstract: | The matrix least squares (LS) problem min
X
|AXB
T
− T|
F
is trivial and its solution can be simply formulated in terms of the generalized inverse of A and B. Its generalized problem min
X
1,X
2|A
1
X
1
B
1
T
+ A
2
X
2
B
2
T
− T|
F
can also be regarded as the constrained LS problem min
X=diag (X
1,X
2)|AXB
T
− T|
F
with A = A
1, A
2] and B = B
1, B
2]. The authors transform T to $
\tilde T
$
\tilde T
such that min
X
1,X
2 |A
1
X
1
B
1
T
+ A
2
X
2
B
2
T
− T|
F
is equivalent to min
X=diag(X
1,X
2) |AXB
T
− $
\tilde T
$
\tilde T
|
F
whose solutions are included in the solution set of unconstrained problem min
X
|AXB
T
− $
\tilde T
$
\tilde T
|
F
. So the general solutions of min
X
1,X
2 |A
1X
1
B
1
T
+ A
2
X
2
B
2
T
− T|
F
are reconstructed by selecting the parameter matrix in that of min
X
|AXB
T
− $
\tilde T
$
\tilde T
|
F
. |
| |
| Keywords: | least squares problem generalized inverse solution set general solutions parameter matrix |
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