Copyright © 2010 Linlin Zhao and Guoliang Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We first consider the following inverse eigenvalue problem: given X∈Cn×m and a diagonal matrix Λ∈Cm×m, find n×n Hermite-Hamilton matrices K and M such that KX=MXΛ. We then consider an optimal approximation problem: given n×n Hermitian matrices Ka and Ma, find a solution (K,M) of the above inverse problem such that ∥K-Ka∥2+∥M-Ma∥2=min⁡. By using the Moore-Penrose generalized inverse and the singular value decompositions, the solvability conditions and the representations of the general solution for the first problem are derived. The expression of the solution to the second problem is presented.