Computing diagonal form and Jacobson normal form of a matrix using Gr\"obner bases

TL;DR

This paper introduces two algorithms using Gr"obner bases to compute diagonal forms of matrices over non-commutative Euclidean domains, facilitating Jacobson and Smith normal form calculations with improved efficiency and precision.

Contribution

The paper presents novel algorithms employing polynomial strategies in Ore localizations for non-commutative matrices, enhancing computational efficiency and accuracy over existing methods.

Findings

Algorithms perform well with moderate coefficient growth.

Implementation shows significant performance improvements.

Method is applicable to a broad class of non-commutative algebras.

Abstract

In this paper we present two algorithms for the computation of a diagonal form of a matrix over non-commutative Euclidean domain over a field with the help of Gr\"obner bases. This can be viewed as the pre-processing for the computation of Jacobson normal form and also used for the computation of Smith normal form in the commutative case. We propose a general framework for handling, among other, operator algebras with rational coefficients. We employ special "polynomial" strategy in Ore localizations of non-commutative $G$-algebras and show its merits. In particular, for a given matrix $M$ we provide an algorithm to compute $U,V$ and $D$ with fraction-free entries such that $UMV=D$ holds. The polynomial approach allows one to obtain more precise information, than the rational one e. g. about singularities of the system. Our implementation of polynomial strategy shows very impressive performance, compared with methods, which directly use fractions. In particular, we experience quite moderate swell of coefficients and obtain uncomplicated transformation matrices. This shows that this method is well suitable for solving nontrivial practical problems. We present an implementation of algorithms in SINGULAR:PLURAL and compare it with other available systems. We leave questions on the algorithmic complexity of this algorithm open, but we stress the practical applicability of the proposed method to a bigger class of non-commutative algebras.