Nearly optimal algorithms for canonical matrix forms

A Las-Vegas-type probabilistic algorithm is presented for finding the Frobenius canonical form of an $n \times n$ matrix $T$ over any field $\mathfrak{K}$. The algorithm requires $O^{\sim}(\operatorname{MM}(n)) = \operatorname{MM}(n) \cdot (\log n)^{O(1)}$ operations in $\mathfrak{K}$, where $O(\operatorname{MM}(n))$ operations in K are sufficient to multiply two $n \times n$ matrices over $\mathfrak{K}$. This nearly matches the lower bound of $\Omega (\operatorname{MM}(n))$ operations in $\mathfrak{K}$ for this problem and improves on the $O(n^{4})$ operations in $\mathfrak{K}$ required by the previous best-known algorithms. A fast parallel implementation of the algorithm is also demonstrated for the Frobenius form, which is processor-efficient on a PRAM. As an application we give an algorithm to evaluate a polynomial $g \in {\mathfrak{K}}[x]$ at $T$ which requires only $O^{\sim}(\operatorname{MM}(n))$ operations in $\mathfrak{K}$ when $\deg g \leq n^{2}$. Other applications include sequential and parallel algorithms for computing the minimal and characteristic polynomials of a matrix, the rational Jordan form of a matrix (for testing whether two matrices are similar), and matrix powering, which are substantially faster than those previously known.