Extension of Cayley-Hamilton theorem for arbitrary polynomial matrices

We give an alternative, more convenient, expression of the Cayley-Hamilton theorem when polynomial matrices of arbitrary degree are involved. Based on the results of the algorithm of Fragulis et al. (1991) for the computation of the inverse of a polynomial matrix, certain relationships among the coefficient matrices of the given polynomial matrix are obtained. We also propose two ways of finding the powers of a polynomial matrix: one in terms of its coefficient matrices and the other making use of the generalized Cayley-Hamilton theorem. These methods are of closed form and are easily implemented in a digital computer.