Nonnegative Eigenvectors of Symmetric Matrices

TL;DR

This paper explores conditions under which symmetric matrices have nonnegative eigenvectors, extending Perron-Frobenius results to matrices with exactly two eigenvalues and examining cases with more eigenvalues.

Contribution

It demonstrates that symmetric matrices with exactly two eigenvalues always have nonnegative eigenvectors, and shows that matrices with more than two eigenvalues may lack such vectors.

Findings

Symmetric matrices with two eigenvalues always have nonnegative eigenvectors.

Matrices with more than two eigenvalues can lack nonnegative eigenvectors.

Extension of Perron-Frobenius theorem to a new class of matrices.

Abstract

For matrices with all nonnegative entries, the Perron-Frobenius theorem guarantees the existence of an eigenvector with all nonnegative components. We show that the existence of such an eigenvector is also guaranteed for a very different class of matrices, namely real symmetric matrices with exactly two eigenvalues. We also prove a partial converse, that among real symmetric matrices with any more than two eigenvalues there exist some having no nonnegative eigenvector.