Least-Squares Solutions of XTAX = B over Positive Semidefinite Matrixes A (Journal of Computational Mathematics, 2003)
This paper is mainly concerned with solving the following two problems:Problem Ⅰ. Given $X\in R^{n× m},B\in R^{m× m}$. Find $A\in P_n$ such that $$\|X^TAX-B\|_F=\min,$$ where $P_n=\{A\in R^{n× n}| x^TAx≥ 0, \forall\,x\in R^n\}$.Problem Ⅱ. Given $\widetilde{A}\in R^{n× n}.$ Find $\widetilde{A}\in S_E$ such that $$\|\widetilde{A}-\hat{A}\|_F=\min_{A\in S_E}\|\widetilde{A}-A\|_F,$$ where $\|·\|_F$ 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.
Keywords: positive semidefinite matrix; Least-square problem; Frobenius norm.