The reduced Cayley–Hamilton equation for a pair of commuting matrices

Let A be n × n matrix of rank r. Then xn−r divides the characteristic polynomial det(xI − A) of A in the ring C[x] of polynomials in x over the complex field C. Let δA(x) = xr−ndet(xI − A). Then δA(A)A = O (Segercrantz (1992) [9]). Let A, B be n × n matrices of rank r and s respectively. If AB = BA, then xn−syn−r divides the polynomial det(xA − yB) in the ring C[x,y] of polynomials in x, y over the complex field. Let δA,B(x, y) = xs−nyr−ndet(xI − A). In this paper, we prove that δA,B(B, A)AB = O under these conditions.