On equivalence classes of dissipative Hamiltonian pencils

2026/06/25 by Maria Dronka
Mathematics · #math.RA #msc:15A18 #msc:15A21 #msc:15A22

paper · pdf

21 pages, 2 figures

arxiv created 2026/06/25 · arxiv updated 2026/07/31

Abstract

We study matrix pencils of the form λE - (J-R)Q, where Q^*E = E^*Q ≥ 0, J^* = -J, R^* = R ≥ 0, and λE - Q is regular, in the framework of Pokrzywa's ordering of strict equivalence orbits. We characterise the maximal elements, analyse some properties of the orbit closures, and derive a restriction on possible degenerations when Q = I. We also consider the case where λE - Q may be singular, which naturally arises in the study of orbit closures, and obtain a partial result on the Kronecker canonical form. The theory is illustrated by numerous examples.

Citations

• Minimal degenerations of orbits of skew-symmetric matrix pencils
• Generic eigenstructures of Hermitian pencils
• Matrix pencils with coefficients that have positive semidefinite Hermitian part
• Optimal control of port-Hamiltonian descriptor systems with minimal energy supply
• A linear relation approach to port-Hamiltonian differential-algebraic equations
• Distance problems for dissipative Hamiltonian systems and related matrix polynomials
• Generic symmetric matrix pencils with bounded rank
• Robust port-Hamiltonian representations of passive systems
• Linear algebra properties of dissipative Hamiltonian descriptor systems
• Port-Hamiltonian descriptor systems
• Generic skew-symmetric matrix polynomials with fixed rank and fixed odd grade

Related

• A constructive proof of Pokrzywa's theorem about perturbations of matrix\n pencils
• Spectral theory of infinite dimensional dissipative Hamiltonian systems
• Linear algebra properties of dissipative Hamiltonian descriptor systems
• Matrix pencils with coefficients that have positive semidefinite Hermitian part
• Minimal degenerations of orbits of skew-symmetric matrix pencils