Efficient computation of the characteristic polynomial of a polynomial matrix

This paper presents an efficient algorithm to compute the characteristic polynomial of a polynomial matrix. We impose the following condition on given polynomial matrix M. Let M-0 be the constant part of M, i.e. M-0 = M (mod (y, ..., z)), where y, ..., z are indeterminates in M. Then, all eigenvalues of M-0 must be distinct. In this case, the minimal polynomial of M and the characteristic polynomial of M agree, i.e. the characteristic polynomial f(x, y, ..., z) = /xE - M/ is the minimal degree (w.r.t. x) polynomial satisfying f(M, y, ..., z) = 0. We use this fact to compute f(x, y, .., z). More concretely, we determine the coefficients of f(x, y, ...,z) little by little with basic matrix operations, which makes the algorithm quite efficient. Numerical experiments are given to compare the algorithm with conventional ones.