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 $\KK$. The algorithm requires $\softO(\MM(n))=\MM(n)\cdot(\log n)^{O(1)}$ operations in $\KK$, where $O(\MM(n))$ operations in $\KK$ are sufficient to multiply two $n\times n$ matrices over $\KK$. This nearly matches the lower bound of $\Omega(\MM(n))$ operations in $\KK$ for this problem, and improves on the $O(n^4)$ operations in $\KK$ required by the previously 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 $g\in\KK[x]$ at $T$ which requires only $\softO(\MM(n))$ operations in $\KK$ 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 for matrix powering which are substantially faster than those previously known.

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