On the Computation of Minimal Polynomial, Cyclic Vectors and the Frobenius Form

Various algorithms connected with the computation of the minimal polynomial of a square n×n matrix over a field k are presented here. The complexity of the first algorithm, where the complete factorization of the characteristic polynomial is needed, is O( √ nn). It produces the minimal polynomial and all characteristic subspaces of a matrix of size n. Furthermore an iterative algorithm for the minimal polynomial is presented with complexity O(n+nm), where m is a parameter of the used Shift-Hessenberg matrix. It does not require knowledge of the characteristic polynomial. Important here is the fact that the average value of m or mA is ≈ log n. Next we are concerned with the topic of finding a cyclic vector first for a matrix whose characteristic polynomial is square-free. Using the Shift-Hessenberg form leads to an algorithm at cost O(n + mn). A more sophisticated recurrent procedure gives the result in O(n) steps. In particular, a normal basis for an extended finite field will be obtained complexity O(n + n log q). Finally the Frobenius form is obtained with asymptotic average complexity O(n log n). All algorithms are deterministic. In all four cases, the complexity obtained is better than for the heretofore best known deterministic algorithm. The results are summarized in Tables 1, 2, 3 and 4.