Fast algorithms for matrix normal forms

A Las Vegas type probabilistic algorithm is presented for computing the Frobenius normal form of an n*n matrix T over any field K. The algorithm requires O/sup approximately /(MM(n))=MM(n)/sup ./(log n)/sup O(1)/ operations in K, where O(MM(n)) operations in K are sufficient to multiply two n*n matrices over K. This nearly matches the lower bound of Omega (MM(n)) operations in K for this problem, and improves on the O(n/sup 4/) operations in K required by the previously best known algorithm. The author applies the algorithm to evaluate a polynominal g in K(x) at T with /sup approximately /(MM(n)) operations in K when deg g>