
D.11.3 jacobson_lib
 Library:
 jacobson.lib
 Purpose:
 Algorithms for Smith and Jacobson Normal Form
 Author:
 Kristina Schindelar, Kristina.Schindelar@math.rwthaachen.de,
Viktor Levandovskyy, levandov@math.rwthaachen.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 noncommutative, 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:
[1] N. Jacobson, 'The theory of rings', AMS, 1943.
[2] 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.
[3] V. Levandovskyy, K. Schindelar 'Computing Jacobson normal form using Groebner bases',
to appear in Journal of Symbolic Computation, 2010.
Procedures:
See also:
control_lib.
