Canonical forms for pairs of matrices associated with Lagrangian and Dirac subspaces

TL;DR

This paper establishes canonical forms for pairs of complex matrices under specific transformations, focusing on cases related to Lagrangian and Dirac subspaces in dissipative Hamiltonian systems.

Contribution

It introduces new canonical forms for matrix pairs associated with Lagrangian and Dirac subspaces under particular transformation groups.

Findings

Derived canonical forms for matrix pairs under specified transformations.

Analyzed special cases involving symmetric and Hermitian matrices.

Connected results to Lagrangian and Dirac subspaces in Hamiltonian systems.

Abstract

We derive the canonical forms for a pair of $n\times n$ complex matrices $(E,Q)$ under transformations $(E,Q) \rightarrow (UEV,U^{-T}QV)$, and $(E,Q) \rightarrow (UEV,U^{-*}QV)$, where $U$ and $V$ are nonsingular complex matrices. We, in particular, consider the special cases of $E^TQ$ and $E^*Q$ being (skew-)symmetric and (skew-)Hermitian, respectively, that are associated with Lagrangian and Dirac subspaces and related linear-time invariant dissipative Hamiltonian descriptor systems.