# Singular          ### D.11.3 jacobson_lib

Library:
jacobson.lib
Purpose:
Algorithms for Smith and Jacobson Normal Form
Author:
Kristina Schindelar, Kristina.Schindelar@math.rwth-aachen.de,
Viktor Levandovskyy, levandov@math.rwth-aachen.de

Overview:
We work over a ring R, that is an Euclidean principal ideal domain. If R is commutative, we suppose R to be a polynomial ring in one variable. If R is non-commutative, we suppose R to have two variables, say x and d. We treat then the basering as the Ore localization of R with respect to the mult. closed set S = K[x] without 0. Thus, we treat basering as principal ideal ring with d a polynomial variable and x an invertible one.
Note, that in computations no division by x will actually happen.

Given a rectangular matrix M over R, one can compute unimodular (that is invertible) square matrices U and V, such that U*M*V=D is a diagonal matrix. Depending on the ring, the diagonal entries of D have certain properties.

We call a square matrix D as before 'a weak Jacobson normal form of M'. It is known, that over the first rational Weyl algebra K(x)<d>, D can be further transformed into a diagonal matrix (1,1,...,1,f,0,..,0), where f is in K(x)<d>. We call such a form of D the strong Jacobson normal form. The existence of strong form in not guaranteed if one works with algebra, which is not rational Weyl algebra.

References:

 N. Jacobson, 'The theory of rings', AMS, 1943.
 Manuel Avelino Insua Hermo, 'Varias perspectives sobre las bases de Groebner :
Forma normal de Smith, Algorithme de Berlekamp y algebras de Leibniz'.
PhD thesis, Universidad de Santiago de Compostela, 2005.
 V. Levandovskyy, K. Schindelar 'Computing Jacobson normal form using Groebner bases',
to appear in Journal of Symbolic Computation, 2010.

Procedures:

 D.11.3.1 smith compute the Smith Normal Form of M over commutative ring D.11.3.2 jacobson compute a weak Jacobson Normal Form of M over non-commutative ring D.11.3.3 divideUnits create ones out of units in the output of smith or jacobson 