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^~(MM(n))=MM(n)^.(log n)^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 Ω(MM(n)) operations in K for this problem, and improves on the O(n ⁴) operations in K required by the previously best known algorithm. The author applies the algorithm to evaluate a polynominal g∈K[x] at T with ^~(MM(n)) operations in K when deg g⩽n². This nearly matches a lower bound of Ω(MM(n)) operations in K when deg g⩽2. Other applications include algorithms for computing the minimal polynomial of a matrix, the rational Jordan form of a matrix, for testing whether two matrices are similar, and for matrix powering, which are substantially faster than those previously known