Frobenius form in expected matrix multiplication time over sufficiently large fields

A new randomized algorithm is presented for computing the Frobenius form of an n×n matrix over a sufficiently large field K. Let 2 < ω ≤ 3 be such that two matrices in Kn×n can be multiplied together with O(n) operations in K. If #K ≥ 2n, the new algorithm uses an expected number of O(n) operations in K to compute the Frobenius form of A, matching the lower bound for the cost of this problem. A similarity transformation matrix can be computed with an additional O(n log logn) operations in K. For comparison, the randomized algorithms of Giesbrecht (1993, 1995) and Eberly (2000) use an expected number of O(n logn) operations in K to compute both the form and a similarity transformation matrix. Eberly’s algorithm has the advantage of being applicable over all fields.