Invariant and polynomial identities for higher rank matrices

TL;DR

This paper develops explicit formulas for discriminants, determinants, and characteristic polynomials of higher rank matrices, extending classical matrix identities to tensors of higher order.

Contribution

It introduces permutation tensors and constructs discriminants and determinants for higher rank matrices, generalizing key matrix identities.

Findings

Explicit expressions for discriminants and determinants of higher rank matrices.

Derivation of characteristic polynomials and Cayley-Hamilton theorem for tensors.

Framework for polynomial identities in higher-dimensional matrix-like objects.

Abstract

We exhibit explicit expressions, in terms of components, of discriminants, determinants, characteristic polynomials and polynomial identities for matrices of higher rank. We define permutation tensors and in term of them we construct discriminants and the determinant as the discriminant of order $d$, where $d$ is the dimension of the matrix. The characteristic polynomials and the Cayley--Hamilton theorem for higher rank matrices are obtained there from.