In t
his paper we present an algorit
hm for t
he computation of a diagonal form of a matrix over non-commutative Euclidean domain over a field wit
h t
he
help of Gr
xf6;bner bases. We propose a general framework of Ore localizations of non-commutative G-algebras and s
how its merits and constructiveness. It allows us to
handle, among ot
hers, common operator algebras wit
h 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 fraction-free 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 non-commutative algebras.