Estimation of Structured Distances to Singularity for Matrix Pencils with Symmetry Structures: A Linear Algebra--Based Approach

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. present purely linear algebra-based approach derive explicit computable formulas nearest $(A-\Delta_A)+s(E-\Delta_E)$ such that $A-\Delta_A$ $E-\Delta_E$ have common null vector. then obtain family of lower bounds unstructured distances singularity. Numerical experiments suggest in many cases, there is significant difference between distances. This extends polynomials with higher degrees.