Jordan Decomposition of Non-Hermitian Fermionic Quadratic Forms

TL;DR

This paper provides a rigorous proof of a conjecture regarding the Jordan decomposition of quadratic fermionic Liouvillians, detailing the structure of nilpotent parts and their relation to q-binomial coefficients.

Contribution

It offers a formal proof of a conjecture on the nilpotent Jordan form of fermionic quadratic forms and describes how to compute the Jordan blocks using q-binomial coefficients.

Findings

Proved the conjecture on the nilpotent part of the Jordan decomposition.

Expressed the number of Jordan blocks using q-binomial coefficients.

Outlined a procedure to obtain the Jordan canonical form.

Abstract

We give a rigorous proof of Conjecture 3.1 by Prosen [Prosen T 2010 J. Stat. Mech. $\textbf{2010}$ P07020] on the nilpotent part of the Jordan decomposition of a quadratic fermionic Liouvillian. We also show that the number of the Jordan blocks of each size can be expressed in terms of the coefficients of a polynomial called the $q$-binomial coefficient and describe the procedure to obtain the Jordan canonical form of the nilpotent part.