Details:
Title  Computing diagonal form and Jacobson normal form of a matrix using Gröbner bases  Author(s)  Viktor Levandovskyy, Kristina Schindelar  Type  Article in Journal  Abstract  In this paper we present an algorithm for the computation of a diagonal form of a matrix over noncommutative Euclidean domain over a field with the help of Gröbner bases. We propose a general framework of Ore localizations of noncommutative G algebras and show its merits and constructiveness. It allows us to handle, among others, common operator algebras with rational coefficients. We introduce the splitting of the computation of a normal form (like the Jacobson form over simple domain) for matrices over Ore localizations into the diagonalization (the computation of a diagonal form of a matrix) and the normalization (the computation of the normal form of a diagonal matrix). These ideas are also used for the computation of the Smith normal form in the commutative case. We give a special algorithm for the normalization of a diagonal matrix over the rational Weyl algebra and present counterexamples to its idea over rational shift and q Weyl algebras. Our implementation of the algorithm in Singular:Plural relies on the fractionfree polynomial strategy, details of which will be described in the forthcoming article. It shows quite an impressive performance, compared with methods which directly use fractions. In particular, we experience quite a moderate swell of coefficients and obtain uncomplicated transformation matrices. We leave questions on the algorithmic complexity of this algorithm open, but we stress the practical applicability of the proposed method to a large class of noncommutative algebras.  Keywords  Matrix normal form, Noncommutative Gröbner basis, Matrix diagonalization over ring, Jacobson normal form, Ore localization  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S0747717110001781 
Language  English  Journal  Journal of Symbolic Computation  Volume  46  Number  5  Pages  595  608  Year  2011  Note  Groebner Bases and Applications  Edition  0  Translation 
No  Refereed 
No 
