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

Abstract

Let (λ, v) be a known real eigenpair of an n×n 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. Detailed examples are provided. Although nonnegative irreducible matrices are somewhat emphasized, the main results in this paper are valid for any n × n real matrix with n≥3.