Volume 21, Issue 2
Least-Squares Solutions of XTAX = B over Positive Semidefinite Matrixes A

J. Comp. Math., 21 (2003), pp. 167-174.

Published online: 2003-04

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• Abstract

This paper is mainly concerned with solving the following two problems:
Problem Ⅰ. Given $X\in R^{n\times m},B\in R^{m\times m}$. Find $A\in P_n$ such that $$\|X^TAX-B\|_F=\min,$$ where $P_n=\{A\in R^{n\times n}| x^TAx\geq 0, \forall\,x\in R^n\}$.
Problem Ⅱ. Given $\widetilde{A}\in R^{n\times n}.$ Find $\widetilde{A}\in S_E$ such that $$\|\widetilde{A}-\hat{A}\|_F=\min_{A\in S_E}\|\widetilde{A}-A\|_F,$$ where $\|\cdot\|_F$ is Frobenius norm, and $S_E$ denotes the solution set of Problem I.
The general solution of problem I has been given. It is proved that there exists a unique solution for Problem II. The expression of this solution for corresponding Problem II for some special case will be derived.

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@Article{JCM-21-167, author = {Xie , Dong-Xiu and Zhang , Lei}, title = {Least-Squares Solutions of XTAX = B over Positive Semidefinite Matrixes A}, journal = {Journal of Computational Mathematics}, year = {2003}, volume = {21}, number = {2}, pages = {167--174}, abstract = {

This paper is mainly concerned with solving the following two problems:
Problem Ⅰ. Given $X\in R^{n\times m},B\in R^{m\times m}$. Find $A\in P_n$ such that $$\|X^TAX-B\|_F=\min,$$ where $P_n=\{A\in R^{n\times n}| x^TAx\geq 0, \forall\,x\in R^n\}$.
Problem Ⅱ. Given $\widetilde{A}\in R^{n\times n}.$ Find $\widetilde{A}\in S_E$ such that $$\|\widetilde{A}-\hat{A}\|_F=\min_{A\in S_E}\|\widetilde{A}-A\|_F,$$ where $\|\cdot\|_F$ is Frobenius norm, and $S_E$ denotes the solution set of Problem I.
The general solution of problem I has been given. It is proved that there exists a unique solution for Problem II. The expression of this solution for corresponding Problem II for some special case will be derived.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/10269.html} }
TY - JOUR T1 - Least-Squares Solutions of XTAX = B over Positive Semidefinite Matrixes A AU - Xie , Dong-Xiu AU - Zhang , Lei JO - Journal of Computational Mathematics VL - 2 SP - 167 EP - 174 PY - 2003 DA - 2003/04 SN - 21 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/10269.html KW - positive semidefinite matrix, Least-square problem, Frobenius norm. AB -

This paper is mainly concerned with solving the following two problems:
Problem Ⅰ. Given $X\in R^{n\times m},B\in R^{m\times m}$. Find $A\in P_n$ such that $$\|X^TAX-B\|_F=\min,$$ where $P_n=\{A\in R^{n\times n}| x^TAx\geq 0, \forall\,x\in R^n\}$.
Problem Ⅱ. Given $\widetilde{A}\in R^{n\times n}.$ Find $\widetilde{A}\in S_E$ such that $$\|\widetilde{A}-\hat{A}\|_F=\min_{A\in S_E}\|\widetilde{A}-A\|_F,$$ where $\|\cdot\|_F$ is Frobenius norm, and $S_E$ denotes the solution set of Problem I.
The general solution of problem I has been given. It is proved that there exists a unique solution for Problem II. The expression of this solution for corresponding Problem II for some special case will be derived.

Dong-Xiu Xie & Lei Zhang. (1970). Least-Squares Solutions of XTAX = B over Positive Semidefinite Matrixes A. Journal of Computational Mathematics. 21 (2). 167-174. doi:
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