The Cayley-Hamilton theorem for supermatrices

TL;DR

This paper extends the Cayley-Hamilton theorem to supermatrices by defining a characteristic polynomial based on the superdeterminant and proving that supermatrices satisfy their characteristic equations, including special cases and examples.

Contribution

It introduces a new definition of the characteristic polynomial for supermatrices and proves the Cayley-Hamilton theorem in this context, including factorization properties.

Findings

Supermatrices satisfy their characteristic equations.

Construction of lower-degree annihilating polynomials.

Application to particular cases and examples.

Abstract

Starting from the expression for the superdeterminant of (xI-M), where M is an arbitrary supermatrix, we propose a definition for the corresponding characteristic polynomial and we prove that each supermatrix satisfies its characteristic equation. Depending upon the factorization properties of the basic polynomials whose ratio defines the above mentioned determinant we are able to construct polynomials of lower degree which are also shown to be annihilated by the supermatrix. Some particular cases and examples are considered.