Jordan decomposition of non-Hermitian fermionic quadratic forms

Abstract

Abstract We give a rigorous proof of conjecture 3.1 by Prosen (2010 J. Stat. Mech. 2010 P07020) on the nilpotent part of the Jordan decomposition of a quadratic fermionic Liouvillian. We also show that the number of 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 for obtaining the Jordan canonical form of the nilpotent part.