Cayley-Hamilton Theorem for Matrices over an Arbitrary Ring

Abstract

For an n×n matrix A over an arbitrary unitary ring R, we obtain the following Cayley-Hamilton identity with right matrix coefficients: (λ0I+C0)+A(λ1I+C1)+… +An-1(λn-1I+Cn-1)+An (n!I+Cn) = 0, where λ0+λ1x+…+λn-1 xn-1+n!xn is the right characteristic polynomial of A in R[x], I ∈ Mn(R) is the identity matrix and the entries of the n×n matrices Ci, 0 ≤ i ≤ n are in [R,R]. If R is commutative, then C0 = C1 = … = Cn-1 = Cn = 0 and our identity gives the n! times scalar multiple of the classical Cayley-Hamilton identity for A.