Estimation to structured distances to singularity for matrix pencils with symmetry structures: A linear algebra-based 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, $\ast$-even, $\ast$-odd, $\ast$-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 $(A-\hat{I}{”}_A)+s(E-\hat{I}{”}_E)$ such that $A-\hat{I}{”}_A$ and $E-\hat{I}{”}_E$ 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.

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