Abstract

In this paper we study the problem of transforming, via invertible column operations, a matrix polynomial into a variety of shifted forms. Examples of forms covered in our framework include a column reduced form, a triangular form, a Hermite normal form or a Popov normal form along with their shifted counterparts.

By obtaining degree bounds for unimodular multipliers of shifted Popov forms we are able to embed the problem of computing a normal form into one of determining a shifted form of a minimal polynomial basis for an associated matrix polynomial. Shifted minimal polynomial bases can be computed via sigma bases (click here) and in Popov form via Mahler systems (click here). The latter method gives a fraction-free algorithm for computing matrix normal forms.