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[a1,...., an] [ 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. © 2006 Elsevier Ltd. All rights reserved.
Beckermann, B., Labahn, G., & Villard, G. (2006). Normal forms for general polynomial matrices. Journal of Symbolic Computation, 41(6), 708–737. https://doi.org/10.1016/j.jsc.2006.02.001