On the condition numbers of a multiple generalized eigenvalue

TL;DR

This paper extends the understanding of condition numbers for multiple eigenvalues from standard to generalized eigenvalue problems, revealing that multiple eigenvalues can have multiple condition numbers even in Hermitian cases.

Contribution

It characterizes the condition numbers of multiple eigenvalues in generalized problems, showing they can vary and are linked to singular values of eigenvector outer products.

Findings

Multiple eigenvalues have multiple condition numbers in generalized problems.

Condition numbers are expressed via singular values of eigenvector outer products.

Even in Hermitian definite cases, multiple eigenvalues exhibit multiple condition numbers.

Abstract

For standard eigenvalue problems, a closed-form expression for the condition numbers of a multiple eigenvalue is known. In particular, they are uniformly 1 in the Hermitian case, and generally take different values in the non-Hermitian case. We consider the generalized eigenvalue problem and identify the condition numbers of a multiple eigenvalue. Our main result is that a multiple eigenvalue generally has multiple condition numbers, even in the Hermitian definite case. The condition numbers are characterized in terms of the singular values of the outer product of the corresponding left and right eigenvectors.