Matrix pencils with coefficients that have positive semidefinite Hermitian part

TL;DR

This paper investigates matrix pencils with coefficients having positive semidefinite Hermitian parts, analyzing their spectral properties, numerical range, and implications for stability and eigenvalue location, extending the understanding of dissipative Hamiltonian structures.

Contribution

It introduces and thoroughly analyzes matrix pencils with positive semidefinite Hermitian coefficients, relaxing spectral restrictions of dissipative Hamiltonian pencils and providing new spectral and stability conditions.

Findings

Spectral properties are heavily restricted for dissipative Hamiltonian pencils.

Matrix pencils with positive semidefinite Hermitian parts have well-characterized spectral and numerical range properties.

Derived conditions ensure spectra lie in the left half-plane and bounds on the index.

Abstract

We analyze when an arbitrary matrix pencil is equivalent to a dissipative Hamiltonian pencil and show that this heavily restricts the spectral properties. In order to relax the spectral properties, we introduce matrix pencils with coefficients that have positive semidefinite Hermitian parts. We will make a detailed analysis of their spectral properties and their numerical range. In particular, we relate the Kronecker structure of these pencils to that of an underlying skew-Hermitian pencil and discuss their regularity, index, numerical range, and location of eigenvalues. Further, we study matrix polynomials with positive semidefinite Hermitian coefficients and use linearizations with positive semidefinite Hermitian parts to derive sufficient conditions for a spectrum in the left half plane and derive bounds on the index.