The minimal polynomial is often the same as the characteristic polynomial, but not always. For example, if is a multiple of the identity matrix, then its minimal polynomial is since the kernel of is already the entire space; on the other hand its characteristic polynomial is (the only eigenvalue is , and the degree of the characteristic polynomial is always equal to the dimension of the space). The minimal polynomial always divides the characteristic polynomial, which is one way of formulating the Cayley–Hamilton theorem (for the case of matrices over a field).