Least-squares solution for inverse eigenpair problem of nonnegative definite matrices

Suppose we know some eigenvalues lambda(i) and eigenvectors x(i) associated with lambda(i), i = 1, 2,..., m for a positive semidefinite (may be unsymmetric) matrix. Let X = (x(1), x(2),..., x(m)), Lambda = diag (lambda(1), lambda(2),..., lambda(m)). In this paper, we mainly discuss solving the following two problems. PROBLEM I. Given X is an element of R-n x m, Lambda = diag(lambda(1),..., lambda(m)). Find matrices A such that parallel to AX - X Lambda parallel to = min, where A is a positive semidefinite (may be unsymmetric) matrix. PROBLEM II. Given (A) over tilde is an element of R-n x n, find (A) over cap is an element of S-E such that [GRAPHICS] where parallel to . parallel to is Frobenius norm, and S-E denotes the solution set of Problem I. An existence theorem of solution for Problems I and II has been given and proved and the general solutions of Problem I have been derived. Sufficient conditions that prove an explicit solution have been provided. (C) 2000 Elsevier Science Ltd. All rights reserved.