Nonnegative Eigenvectors of Symmetric 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.

• MSC: Primary 15B57
• Secondary 52A20

Acknowledgments

I thank the referees for helpful suggestions, including a better version of the hyperplane separation theorem.