An algorithm for the calculation of the minimal polynomial
Stanisław Białas, Małgorzata Białas · Bulletin of the Polish Academy of Sciences Technical Sciences · 2008
where an 6= 0, is called monic if its leading coefficient an = 1. The monic polynomial ψ(λ) of least degree for which ψ(A) = 0 ∈ Mn is called the minimal polynomial of the matrix A ∈Mn. The properties and the applications of the minimal polynomials in the control theory have been presented in [1, 2]. In this paper the simple algorithm is given for the calculation of the degree and coefficients of the minimal polynomial. For the matrix A = [aij ] ∈Mn we will use the following notations: φ(λ) = det(λI − A) – charecteristic polynomial of the matrix A, ψ(λ) – minimal polynomial of the matrix A, vecA = [a11 a12 . . . a1n a21 a22 . . . a2n . . . an1 an2 . . . ann] , A = I ∈Mn, A = AA (k = 1, 2, . . .), (1) A = [a (k) ij ] (k = 0, 1, 2, . . .), a = vecA (k = 0, 1, 2, . . .), Bk = [a a · · · a] (k = 0, 1, 2, . . .) where a is k + 1-th column of the matrix Bk ∈Mn2,k+1,