On the left and right coefficients of the general Cayley-Hamilton identities for an nxn matrix

TL;DR

This paper explores the structure of Cayley-Hamilton identities for matrices over rings with specific gradings, revealing new relations between coefficients and matrix components.

Contribution

It introduces new identities involving the left and right coefficients of Cayley-Hamilton identities for matrices over graded rings.

Findings

Derived explicit relations between coefficients C(i) and D(i).

Established identities involving matrix A and its symmetric characteristic polynomial.

Extended Cayley-Hamilton identities to rings with Grassmann algebra-like grading.

Abstract

An nxn matrix A over an arbitrary unitary ring R satisfies invariant left and right Cayley-Hamilton identities with matrix coefficients C(i), D(i) having commutator sum entries. If R has a grading similar to the case of Grassmann algebras, then we prove that C(i)-D(i)-AC(i+1)+D(i+1)A=-2p(i+1)A1 for all i, where A1 and p(i+1) are the odd components of A and of the symmetric characteristic polynomial of A, respectively.