A Direct Proof of a Theorem Concerning Singular Hamiltonian Systems

TL;DR

This paper provides a direct proof of a theorem related to singular Hamiltonian systems, utilizing a basis change to decouple difference equations, thereby simplifying the proof process.

Contribution

It introduces a novel direct proof method for a key theorem in singular Hamiltonian systems using a basis change to decouple equations.

Findings

Successful decoupling of difference equations

Simplified proof of the theorem

Implications for related equations in the literature

Abstract

This technical report presents a direct proof of Theorem~1 in [1] and some consequences that also account for (20) in [1]. This direct proof exploits a state space change of basis which replaces the coupled difference equations (10) in [1] with two equivalent difference equations which, instead, are decoupled.