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

Suppose we know some eigenvalues λi and eigenvectors xi associated with λi i = 1, 2, …, m for a positive semidefinite (may be unsymmetric) matrix. Let X = (x1,x2,…,xm, Λ = diag (λ1,λ2,…,λm. In this paper, we mainly discuss solving the following two problems. I. Given X ε Rn × m, Λ = diag(λ1, …, λm). Find matrices A such that ;AX − XΛ ; = min, where A is a positive semidefinite (may be unsymmetric) matrix. II. Given à ε Rn × n, find  ε SE such that , where • is Frobenius norm, and SE 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.