Coalescing points for eigenvalues of banded matrices depending on parameters with application to banded random matrix functions

Abstract

In this work, we develop and implement new numerical methods to locate generic degeneracies (i.e., isolated parameters’ values where the eigenvalues coalesce) of banded matrix valued functions. More precisely, our specific interest is in two classes of problems: (i) symmetric, banded, functions A(x) ∈ ℝ^n×n, smoothly depending on parameters x ∈ Ω ⊂ ℝ² and (ii) Hermitian, banded, functions A(x) ∈ ℂ^n×n, smoothly depending on parameters x ∈Ω⊂ ℝ³. The computational task of detecting coalescing points of banded parameter-dependent matrices is very delicate and challenging and cannot be handled using existing eigenvalues’ continuation approaches. For this reason, we present and justify new techniques that will enable continuing path of eigendecompositions and reliably decide whether or not eigenvalues coalesce, well beyond our ability to numerically distinguish close eigenvalues. As important motivation, and illustration, of our methods, we perform a computational study of the density of coalescing points for random ensembles of banded matrices depending on parameters. Relatively to random matrix models from truncated GOE and GUE ensembles, we will give computational evidence in support of power laws for coalescing points, expressed in terms of the size and bandwidth of the matrices.