A generalized Cayley–Hamilton theorem for polynomials with matrix coefficients

A family F of square matrices of the same order is called a quasi-commuting family if (AB-BA)C=C(AB-BA) for all A,B,C∈F where A,B,C need not be distinct. Let fk(x1,x2,…,xp),(k=1,2,…,r), be polynomials in the indeterminates x1,x2,…,xp with coefficients in the complex field C, and let M1,M2,…,Mr be n×n matrices over C which are not necessarily distinct. Let and let δF(x1,x2,…,xp)=detF(x1,x2,…,xp). In this paper, we prove that, for n×n matrices A1,A2,…,Ap over C, if {A1,A2,…,Ap,M1,M2,…,Mr} is a quasi-commuting family, then F(A1,A2,…,Ap)=O implies that δF(A1,A2,…,Ap)=O.