On equivalence classes of dissipative Hamiltonian pencils

Abstract

We study matrix pencils of the form $λE - (J-R)Q$, where $Q^*E = E^*Q \geq 0$, $J^* = -J$, $R^* = R \geq 0$, 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.