On the Computation of the Minimal Polynomial of a Polynomial Matrix

It is well known from the Cayley Hamilton theorem that every matrix A ∈ Rr×r satisfies its characteristic equation (Gantmacher, 1959), i.e., if p (s) := det (sIr −A) = sr+p1sr−1+· · ·+pr, then p (A) = 0. The Cayley Hamilton theorem is still valid for all cases of matrices over a commutative ring (Atiyah and McDonald, 1964), and thus for multivariable polynomial matrices. Another form of the Cayley-Hamilton theorem, also known as the relative Caley-Hamilton theorem, is given in terms of the fundamental matrix sequence of the resolvent of the matrix, i.e., if (sIr −A)−1 = ∑∞ i=0 Φis −i then Φk+p1Φk−1+ · · · + prΦk−r = 0. The Cayley-Hamilton theorem was investigated for the matrix pencil case A (s) = A0+A1s in (Mertzios and Christodoulou, 1986), and the respective relative Cayley-Hamilton theorem in (Lewis, 1986). The Cayley-Hamilton theorem was extended to matrix polynomials (Fragulis, 1995; Kitamoto, 1999; Yu and Kitamoto, 2000), to standard and singular bivariate matrix pencils (Givone and Roesser, 1973; Ciftcibaci and Yuksel, 1982; Kaczorek, 1995a; 1989; Vilfan, 1973), M-D matrix pencils in (Gałkowski, 1996; Theodorou, 1989) and n-d polynomial matrices (Kaczorek, 2005). The Cayley-Hamilton theorem was also extended to non-square matrices, nonsquare block matrices and singular 2D linear systems with non-square matrices (Kaczorek, 1995b; 1995c; 1995d). The reason behind the interest in the Caley-Hamilton theorem is its applications in control systems, i.e., the calculation of controllability and observability grammians and the state-transition matrix, electrical circuits, systems with delays, singular systems, 2-D linear systems, the calculation of the powers of matrices and inverses, etc. Of particular importance for the determination of the characteristic polynomial of a polynomial matrix A (s) = A0 + A1s+ · · ·+Aqsq ∈ Rr×r[s] are: (a) the FaddeevLeverrier algorithm (Faddeev and Faddeeva, 1963; Helmberg et al., 1993) which is fraction free and needs r(r − 1) polynomial multiplications, (b) the CHTB method presented in (Kitamoto, 1999), which needs r (q + 1) polynomial multiplications (its shortcomings are that it cannot be used for a polynomial matrix A (s) when A0 has multiple eigenvalues, and it needs to compute first the eigenvalues and eigenvectors of A0), and (c) the CHACM method presented in (Yu and Kitamoto, 2000), which needs 7 12r +O ( r ) polynomialmultiplications (a CHTB method given with an artifical constant matrix in order to release the restrictions of the CHTB method, which needs no condition on the given matrix, does not have to solve any eigenvalue problem and is fraction free). Except for the characteristic polynomial of a constant matrix, say p (s), with the nice property p (A) = 0, there is also another polynomial, known as the minimal polynomial, say m (s), which is the least degree monic polynomial that satisfies the equation m (A) = 0 (Gantmacher, 1959). Since the minimal polynomial has a lower degree than the characteristic polynomial, it helps us to solve faster problems such as the computation of the inverse or power of a matrix.