The number of distinct eigenvalues of a matrix after perturbation

TL;DR

This paper establishes a theorem linking the change in the number of distinct eigenvalues of a matrix after perturbation to the original eigenvalues, the rank of the update, and the matrix's nondiagonalizability, with implications for Krylov methods.

Contribution

The paper introduces a new theorem that bounds the increase in distinct eigenvalues after perturbation, applicable to both symmetric and nonsymmetric matrices, including rank one updates.

Findings

A rank one update to a diagonalizable matrix can at most double the number of distinct eigenvalues.

The theorem applies to matrices of arbitrary size and perturbation magnitude.

The number of Krylov iterations needed to solve a linear system can at most double after a rank one update.

Abstract

We prove a new theorem relating the number of distinct eigenvalues of a matrix after perturbation to the prior number of distinct eigenvalues, the rank of the update, and the degree of nondiagonalizability of the matrix. In particular, a rank one update applied to a diagonalizable matrix can at most double the number of distinct eigenvalues. The theorem applies to both symmetric and nonsymmetric matrices and perturbations, of arbitrary magnitudes. An an application, we prove that in exact arithmetic the number of Krylov iterations required to exactly solve a linear system involving a diagonalizable matrix can at most double after a rank one update.