Inclusion regions and bounds for the eigenvalues of matrices with a known eigenpair

TL;DR

This paper develops methods to locate and bound the eigenvalues of a real matrix using a known eigenpair, extending Gershgorin's theorem with new regions and bounds applicable to general matrices.

Contribution

It introduces eigenvalue inclusion regions based on known eigenpairs, utilizing a union of Gershgorin discs, and provides bounds for the largest eigenvalue in absolute value, applicable to all real matrices.

Findings

Eigenvalue regions are unions of Gershgorin discs of the second type.

Upper bounds for the largest eigenvalue are derived.

Results are applicable to any real square matrix, with emphasis on nonnegative irreducible matrices.

Abstract

Let ({\lambda}, v) be a known real eigenpair of a square real matrix A. In this paper it is shown how to locate the other eigenvalues of A in terms of the components of v. The obtained region is a union of Gershgorin discs of the second type recently introduced by the authors in a previous paper. Two cases are considered depending on whether or not some of the components of v are equal to zero. Upper bounds are obtained, in two different ways, for the largest eigenvalue in absolute value of A other than {\lambda}. Detailed examples are provided. Although nonnegative irreducible matrices are somewhat emphasized, the main results in this paper are valid for any square real matrix.