Eigenvalue Paths Arising From Matrix Paths

TL;DR

This paper investigates the continuous paths of eigenvalues of matrices as their entries vary smoothly, analyzing their behavior under perturbations and their relation to matrix pairings, with applications to polynomial analogs.

Contribution

It provides new insights into the behavior of eigenvalue paths under matrix perturbations and establishes analogs for polynomial eigenvalues.

Findings

Eigenvalues follow continuous paths under matrix entry variations.

Eigenpaths exhibit specific behaviors under small perturbations.

Results extend to monic polynomial eigenvalues.

Abstract

It is known (see e.g. [2], [4], [5], [6]) that continuous variations in the entries of a complex square matrix induce continuous variations in its eigenvalues. If such a variation arises from one real parameter $\alpha \in [0, 1]$, then the eigenvalues follow continuous paths in the complex plane as ${\alpha}$ shifts from $0$ to $1$. The intent here is to study the nature of these eigenpaths, including their behavior under small perturbations of the matrix variations, as well as the resulting eigenpairings of the matrices that occur at ${\alpha} = 0$ and ${\alpha} = 1$. We also give analogs of our results in the setting of monic polynomials.