The Relation between the Mandelstam and the Cayley-Hamilton Identities

TL;DR

This paper demonstrates the equivalence between Mandelstam and Cayley-Hamilton identities for matrices and extends the approach to supermatrices, linking characteristic polynomials to these identities.

Contribution

It provides a combinatorial derivation of Mandelstam identities from the characteristic polynomial and extends this to supermatrices, establishing their equivalence.

Findings

Mandelstam and Cayley-Hamilton identities are equivalent for matrices.

The method extends to supermatrices, deriving Mandelstam identities from characteristic equations.

The approach offers a unified combinatorial framework for these identities.

Abstract

Starting from the characteristic polynomial for ordinary matrices we give a combinatorial deduction of the Mandelstam identities and viceversa, thus showing that the two sets of relations are equivalent. We are able to extend this construction to supermatrices in such a way that we obtain the Mandelstam identities in this case, once the corresponding characteristic equation is known.