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

2021/05/28 by Anshul Prajapati, Punit Sharma, Prajapati, Anshul +1
Computer Science · Mathematics · Physics and Astronomy · #Matrix Theory and Algorithms #Mathematical functions and polynomials #Quantum Mechanics and Non-Hermitian Physics

paper · pdf · doi:10.48550/arxiv.2105.13656

Abstract

We study the structured distance to singularity for a given regular matrix pencil A+sE, where (A,E)∈ \mathbb S ⊆ (\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-Δ_A)+s(E-Δ_E) such that A-Δ_A and E-Δ_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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