Least-Squares Solutions of <em>X<sup>T</sup>AX = B</em> over Positive Semidefinite Matrixes <em>A</em>
Year: 2003
Author: Dong-Xiu Xie, Lei Zhang
Journal of Computational Mathematics, Vol. 21 (2003), Iss. 2 : pp. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2003-JCM-10269
Journal of Computational Mathematics, Vol. 21 (2003), Iss. 2 : pp. 167–174
Published online: 2003-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 8
Keywords: positive semidefinite matrix Least-square problem Frobenius norm.