Eigenvalue avoidance of structured matrices depending smoothly on a real parameter

TL;DR

This paper investigates eigenvalue avoidance phenomena in structured matrices depending smoothly on a real parameter, using differential geometry to analyze generic behaviors across various matrix classes.

Contribution

It extends the understanding of eigenvalue avoidance from symmetric and Hermitian matrices to a broad range of structured matrices using a geometric approach.

Findings

Eigenvalue avoidance always occurs in some matrix structures.

In other structures, avoidance depends on matrix size, eigenvalue multiplicity, and determinant.

The methods also apply to singular value avoidance in unstructured matrices.

Abstract

We explore the concept of eigenvalue avoidance, which is well understood for real symmetric and Hermitian matrices, for other classes of structured matrices. We adopt a differential geometric perspective and study the generic behaviour of the eigenvalues of regular and injective curves $t \in ]a,b[ \mapsto A(t) \in \mathcal{N} $ where $\mathcal{N}$ is a smooth real Riemannian submanifold of either $\mathbb{R}^{n \times n}$ or $\mathbb{C}^{n \times n}$. We focus on the case where $\mathcal{N}$ corresponds to some class of (real or complex) structured matrices including skew-symmetric, skew-Hermitian, orthogonal, unitary, banded symmetric, banded Hermitian, banded skew-symmetric, and banded skew-Hermitian. We argue that for some structures eigenvalue avoidance always happens, whereas for other structures this may depend on the parity of the size, on the numerical value of the multiple eigenvalue, and possibly on the value of determinant. As a further application of our tools we also study singular value avoidance for unstructured matrices in $\mathbb{R}^{m \times n}$ or $\mathbb{C}^{m \times n}$.