Volume 21, Issue 4
The Inverse Problem for Part Symmetric Matrices on a Subspace

J. Comp. Math., 21 (2003), pp. 505-512.

Published online: 2003-08

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• Abstract

In this paper, the following two problems are considered:
Problem Ⅰ. Given $S \in R^{n×p}, X, B \in R^{n×m}$, find $A \in SR_{s,n}$ such that $AX=B$, where $SR_{s,n}={A \in R^{n×n} | x^T(A-A^T)=0, \ {\rm for} \ {\rm all} \ x \in R(S)}$.

Problem Ⅱ. Given $A^* \in R^{n×n}$, find $\hat{A} \in S_E$ such that $\|\hat{A} -A^*\|={\rm min}_{A \in S_E} \|A-A*\|$, where $S_E$ is the solution set of Problem Ⅰ.

Then necessary and sufficient conditions for the solvability of and the general from of the solutions of problem Ⅰ are given. For problem Ⅱ, the expression for the solution, a numerical algorithm and a numerical example are provided.

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@Article{JCM-21-505, author = {Peng , Zhen-YunHu , Xi-Yan and Zhang , Lei}, title = {The Inverse Problem for Part Symmetric Matrices on a Subspace}, journal = {Journal of Computational Mathematics}, year = {2003}, volume = {21}, number = {4}, pages = {505--512}, abstract = {

In this paper, the following two problems are considered:
Problem Ⅰ. Given $S \in R^{n×p}, X, B \in R^{n×m}$, find $A \in SR_{s,n}$ such that $AX=B$, where $SR_{s,n}={A \in R^{n×n} | x^T(A-A^T)=0, \ {\rm for} \ {\rm all} \ x \in R(S)}$.

Problem Ⅱ. Given $A^* \in R^{n×n}$, find $\hat{A} \in S_E$ such that $\|\hat{A} -A^*\|={\rm min}_{A \in S_E} \|A-A*\|$, where $S_E$ is the solution set of Problem Ⅰ.

Then necessary and sufficient conditions for the solvability of and the general from of the solutions of problem Ⅰ are given. For problem Ⅱ, the expression for the solution, a numerical algorithm and a numerical example are provided.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/10254.html} }
TY - JOUR T1 - The Inverse Problem for Part Symmetric Matrices on a Subspace AU - Peng , Zhen-Yun AU - Hu , Xi-Yan AU - Zhang , Lei JO - Journal of Computational Mathematics VL - 4 SP - 505 EP - 512 PY - 2003 DA - 2003/08 SN - 21 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/10254.html KW - Part symmetric matrix, Inverse problem, Optimal approximation. AB -

In this paper, the following two problems are considered:
Problem Ⅰ. Given $S \in R^{n×p}, X, B \in R^{n×m}$, find $A \in SR_{s,n}$ such that $AX=B$, where $SR_{s,n}={A \in R^{n×n} | x^T(A-A^T)=0, \ {\rm for} \ {\rm all} \ x \in R(S)}$.

Problem Ⅱ. Given $A^* \in R^{n×n}$, find $\hat{A} \in S_E$ such that $\|\hat{A} -A^*\|={\rm min}_{A \in S_E} \|A-A*\|$, where $S_E$ is the solution set of Problem Ⅰ.

Then necessary and sufficient conditions for the solvability of and the general from of the solutions of problem Ⅰ are given. For problem Ⅱ, the expression for the solution, a numerical algorithm and a numerical example are provided.

Zhen-Yun Peng, Xi-Yan Hu & Lei Zhang. (1970). The Inverse Problem for Part Symmetric Matrices on a Subspace. Journal of Computational Mathematics. 21 (4). 505-512. doi:
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