Decompositions and coalescing eigenvalues of symmetric definite pencils depending on parameters

TL;DR

This paper investigates the smoothness and behavior of eigenvalues in symmetric positive definite matrix pencils depending on two parameters, providing theoretical insights and numerical analysis of eigenvalue coalescence in engineering-relevant random matrix models.

Contribution

It offers new theoretical results on eigenvalue smoothness and a numerical study of coalescing eigenvalues in parameter-dependent symmetric pencils.

Findings

Eigenvalues may not be smooth at conical intersections.

Eigenvalue coalescence depends on matrix structure and bandwidth.

Statistical properties of eigenvalue coalescence are characterized in random matrix models.

Abstract

In this work, we consider symmetric positive definite pencils depending on two parameters. That is, we are concerned with the generalized eigenvalue problem $A(x)-\lambda B(x)$, where $A$ and $B$ are symmetric matrix valued functions in ${\mathbb R}^{n\times n}$, smoothly depending on parameters $x\in \Omega\subset {\mathbb R}^2$; further, $B$ is also positive definite. In general, the eigenvalues of this multiparameter problem will not be smooth, the lack of smoothness resulting from eigenvalues being equal at some parameter values (conical intersections). We first give general theoretical results on the smoothness of eigenvalues and eigenvectors for the present generalized eigenvalue problem, and hence for the corresponding projections, and then perform a numerical study of the statistical properties of coalescing eigenvalues for pencils where $A$ and $B$ are either full or banded, for several bandwidths. Our numerical study will be performed with respect to a random matrix ensemble which respects the underlying engineering problems motivating our study.