A fast numerical algorithm for constructing nonnegative matrices with prescribed real eigenvalues

TL;DR

This paper introduces a fast numerical algorithm based on induction principles to construct nonnegative and symmetric nonnegative matrices with prescribed real eigenvalues, facilitating solutions for inverse eigenvalue problems.

Contribution

It provides a novel, efficient numerical method for inverse eigenvalue problems for nonnegative and symmetric nonnegative matrices, including stochastic matrices.

Findings

Algorithm successfully constructs matrices with desired spectra

Applicable to larger-sized problems efficiently

Offers conditions and methods for inverse eigenvalue problems

Abstract

The study of solving the inverse eigenvalue problem for nonnegative matrices has been around for decades. It is clear that an inverse eigenvalue problem is trivial if the desirable matrix is not restricted to a certain structure. Provided with the real spectrum, this paper presents a numerical procedure, based on the induction principle, to solve two kinds of inverse eigenvalue problems, one for nonnegative matrices and another for symmetric nonnegative matrices. As an immediate application, our approach can offer not only the sufficient condition for solving inverse eigenvalue problems for nonnegative or symmetric nonnegative matrices, but also a quick numerical way to solve inverse eigenvalue problem for stochastic matrices. Numerical examples are presented for problems of relatively larger size.