Cayley Hamilton theorem with sandwich coefficients for n$\times$n matrices over a ring satisfying [x,y][u,v]=0

TL;DR

This paper establishes a Cayley-Hamilton theorem with sandwich coefficients for n×n matrices over rings satisfying a specific polynomial identity, expanding classical matrix theory to a broader algebraic context.

Contribution

It introduces a new form of the Cayley-Hamilton identity with sandwich coefficients for matrices over rings satisfying [x,y][u,v]=0, generalizing classical results.

Findings

Cayley-Hamilton identity with sandwich coefficients proven for such rings

The identity involves coefficients c_{i,j} in R with c_{n,n}=(n!)^2

The result applies to matrices over rings satisfying the polynomial identity

Abstract

If A is an n \times n matrix over a ring R satisfying the polynomial identity [x,y][u,v]=0, then an invariant Cayley-Hamilton identity of the form \Sigma A^{i}c_{i,j}A^{j}=0 with c_{i,j}\in R and c_{n,n}=(n!)^2 holds for A.