Fast Parallel Algorithms for Matrix Reduction to Normal Forms

Abstract

We investigate fast parallel algorithms to compute normal forms of matrices and the corresponding transformations. Given a matrix B in M n,n (K), where K is an arbitrary commutative field, we establish that computing a sim- ilarity transformation P such that FP~1BP is in Frobenius normal form can be done in NC2 . Using a reduction to this first problem, a similar fact is then proved for the Smith normal form S(x) of a polynomial matrix A(x )i n M n , m ( K ( x)); to compute unimodular matrices (x) and »(x) such that S(x)(x)A(x)»(x) can be done in NC2 . We get that over concrete fields such as the rationals, these problems are in NC2. Using our previous results we have thus established that the problems of computing transformations over a field extension for the Jordan normal form, and transformations over the input field for the Frobenius and the Smith normal form are all in NC2 . As a corollary we establish a polynomial-time sequential algorithm to compute transformations for the Smith form over K(x).