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

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.