Computing Minimal Polynomials of Matrices

We present and analyse a Monte-Carlo algorithm to computethe minimal polynomial of an n × n matrix over a finite fieldthat requires O(n3) field operations and O(n) random vectors,and is well suited for successful practical implementation.The algorithm, and its complexity analysis, use standard algorithmsfor polynomial and matrix operations. We comparefeatures of the algorithm with several other algorithms in theliterature. In addition we present a deterministic verificationprocedure which is similarly efficient in most cases but has aworst-case complexity of O(n4). Finally, we report the resultsof practical experiments with an implementation of our algorithmsin comparison with the current algorithms in the GAPlibrary.