Estimation to structured distances to singularity for matrix pencils with symmetry structures: A linear algebra-based approach

TL;DR

This paper introduces a linear algebra-based method to compute the structured distance to singularity for various symmetry-structured matrix pencils, providing explicit formulas and bounds, and demonstrating differences between structured and unstructured distances.

Contribution

It develops explicit formulas and bounds for the structured distance to singularity for symmetry-structured matrix pencils, extending to higher-degree polynomials.

Findings

Structured distances can significantly differ from unstructured ones.

Explicit formulas enable efficient computation of structured distances.

Numerical experiments validate the effectiveness of the proposed approach.

Abstract

We study the structured distance to singularity for a given regular matrix pencil $A+sE$, where $(A,E)\in \mathbb S \subseteq (\mathbb C^{n,n})^2$. This includes Hermitian, skew-Hermitian, $*$-even, $*$-odd, $*$-palindromic, T-palindromic, and dissipative Hamiltonian pencils. We present a purely linear algebra-based approach to derive explicit computable formulas for the distance to the nearest structured pencil $(A-\Delta_A)+s(E-\Delta_E)$ such that $A-\Delta_A$ and $E-\Delta_E$ have a common null vector. We then obtain a family of computable lower bounds for the unstructured and structured distances to singularity. Numerical experiments suggest that in many cases, there is a significant difference between structured and unstructured distances. This approach extends to structured matrix polynomials with higher degrees.