Normal forms for general polynomial matrices

  • Authors:

  • Bernhard Beckermann;George Labahn;Gilles Villard

  • Affiliations:

  • Laboratoire Painlevé UMR 8524 (ANO-EDP), Université des Sciences et Technologies de Lille, 59655 Villeneuve dAscq Cedex, France;Symbolic Computation Group, School of Computer Science, University of Waterloo, Waterloo, Ontario, Canada;CNRS-LIP Ecole Normale Supérieure de Lyon, 46, Allée dItalie, 69364 Lyon Cedex 07, France

  • Venue:
  • Journal of Symbolic Computation
  • Year:
  • 2006

Quantified Score

Hi-index 0.00

Visualization

Abstract

We present an algorithm for the computation of a shifted Popov normal form of a rectangular polynomial matrix. For specific input shifts, we obtain methods for computing the matrix greatest common divisor of two matrix polynomials (in normal form) and procedures for such polynomial normal form computations as those of the classical Popov form and the Hermite normal form. The method involves embedding the problem of computing shifted forms into one of computing matrix rational approximants. This has the advantage of allowing for fraction-free computations over integral domains such as Z[z] and K[a"1,...,a"n][z]. In the case of rectangular matrix input, the corresponding multipliers for the shifted forms are not unique. We use the concept of minimal matrix approximants to introduce a notion of minimal multipliers and show how such multipliers are computed by our methods.