An extension of the Cayley-Hamilton theorem to the case of 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 lower-degree polynomials.

Contribution

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

Findings

Supermatrices satisfy their characteristic equations.

Lower-degree annihilating polynomials can be constructed.

The superdeterminant-based characteristic polynomial is well-defined.

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 superdeterminant we are able to construct polynomials of lower degree which are also shown to be annihilated by the supermatrix.