Multiplicities of the structured pseudoeigenvalues

TL;DR

This paper proves that the sum of eigenvalue multiplicities within each connected component of a matrix's structured pseudospectrum remains invariant, providing a detailed and elementary proof of this known fact.

Contribution

It offers an elementary, detailed proof of the invariance of eigenvalue multiplicity sums in structured pseudospectra, clarifying a less emphasized aspect.

Findings

Sum of eigenvalue multiplicities in each connected component is invariant

Provides an elementary proof of a known result

Clarifies the properties of structured pseudospectra

Abstract

The structured pseudospectra of a matrix A are sets of complex numbers that are eigenvalues of matrices X which are near to A and have the same entries as A at a fixed set of places. The sum of multiplicities of the eigenvalues of X inside each connected component of the structured pseudospectra of A does not depend on X. This fact is known, but not so much as it should be. For this reason, we give here an elementary and detailed proof of the result.