Least-Squares Solutions of XTAX = B over Positive Semidefinite Matrixes A

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.