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

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