对称非负定阵一类逆特征值问题

A CLASS OF INVERSE EIGENVALUE PROBLEM FOR SYMMETRIC NONNEGATIVE DEFINITE MATRICES

本文考虑下面两个问题: 问题Ⅰ:给定求使,其中表示Frobenius范数, 问题Ⅱ:给定求使,其中S_E是问题Ⅰ的解集合, 问题Ⅰ、Ⅱ解的存在性和同题Ⅱ解的唯一性已被证明,当A≥0,给出了S_E的通式和A_(LS)的表达式。

In this paper,the following two problems are considered:ProblemA:Given X∈R^(n×m),ΛA=diag(λ_1,…,λ_m),find A∈S such that ‖AX—XA‖=min,where‖·‖is Frobenius norm,S={A∈R^(n×n)|A=A^T,x^TAx≥0,(?)x∈R^n}.Problem B:Given A*∈R^(n×n),find A_(LS)∈S_E such that ‖A~*—A_(LS)‖=(?)‖A~*—A‖,where S_E is the solution set of Problim A.The existence of the solution to Problem A,B and the umigueness of the solution to Problem B are proved. When A≥0,the general form of S_E is given and the expression of A_(LS) is provided.