Fast Parallel Algorithms for Matrix Reduction to Normal Forms

Abstract

Abstract. We investigate fast parallel algorithms to compute normal forms of matrices and the corresponding transformations. Given a matrix B in M L L (K), where K is an arbitrary commutative field, we establish that computing a similarity transformation P such that F"P\ BP is in Frobenius normal form can be done in NC ) . Using a reduction to this first problem, a similar fact is then proved for the Smith normal form S(x) of a polynomial matrix to compute unimodular matrices º(x) and »(x) such that S(x)"º(x)A(x)»(x) can be done in NC) . We get that over concrete fields such as the rationals, these problems are in NC .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 NC ) . As a corollary we establish a polynomial-time sequential algorithm to compute transformations for the Smith form over K [x].

Bivariate polynomial reduction and elimination ideal over finite fields

“…3.10). Such a formalism has occasionally been used to efficiently compute general matrix normal forms Villard (1997);Storjohann (2000).…”

Orbits of monomials and factorization into products of linear forms

“…Once we know the eigenvalues of M and their multiplicities, the last step is the computation of a transition matrix T such that T −1 M T is diagonal. For this step we refer to [19,20,39]. These papers consider the more general problem of computing symbolic representations of the Jordan normal form.…”

A local construction of the Smith normal form of a matrix polynomial

“…A fundamental theorem in matrix theory [5,7] states that two square matrices A and B over a field K are similar if and only if their characteristic matrix polynomials λI − A and λI − B have the same Smith form D(λ). Other applications of this canonical form include finding the Frobenius form [22,24] of a matrix A over a field by computing the invariant factors of the matrix pencil λI − A.…”