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

TL;DR

This work develops canonical forms for pairs of $n\times n$ complex matrices $(E,Q)$ under two transformation groups motivated by energy-based and descriptor-system modeling. It treats both the nonsingular and singular cases, with special attention to structured products where $E^{T}Q$ is symmetric (or skew-symmetric) and $E^{*}Q$ is Hermitian (or skew-Hermitian), which connect to Lagrangian and Dirac subspaces in port-Hamiltonian systems. The authors establish explicit canonical forms, prove uniqueness up to summand permutation, and compute the codimensions of the corresponding orbits under $^{T}$- and $^{*}$-equivalence, using block-structure analyses and interaction dimensions. These results illuminate the algebraic structure of pencils like $\lambda E - Q$ and $\lambda E - LQ$, with potential extensions to broader singular regimes, versal deformations, and orbit stratifications for deeper geometric understanding.

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.

Theorems & Definitions (46)

• Theorem 2.1
• Theorem 2.2
• Definition 2.3: Definition 2 in DeDo11
• Definition 2.4: Definition 3.1 in DeDo11_1
• Theorem 2.5
• Theorem 2.6
• Lemma 3.1
• proof
• Theorem 3.2: Canonical form under $^{T}$-equivalence
• proof