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D.11.3.2 jacobson
Procedure from library jacobson.lib (see jacobson_lib).
- Usage:
- jacobson(M, eng); M matrix, eng an optional int
- Return:
- list
- Assume:
- Basering is a (non-commutative) ring in two variables.
- Purpose:
- compute a weak Jacobson normal form of M over the basering
- Theory:
- Groebner bases and involutions are used, following [3]
- Note:
- A list L of matrices {U,D,V} is returned. That is L[1]*M*L[3]=L[2],
where L[2] is a diagonal matrix and
L[1], L[3] are square invertible polynomial (unimodular) matrices.
Note, that M can be rectangular.
The optional integer eng2 determines the Groebner basis engine:
0 (default) ensures the use of 'slimgb' , otherwise 'std' is used.
- Display:
- If
printlevel =1, progress debug messages will be printed,
if printlevel >=2, all the debug messages will be printed.
Example:
| LIB "jacobson.lib";
ring r = 0,(x,d),Dp;
def R = nc_algebra(1,1); setring R; // the 1st Weyl algebra
matrix m[2][2] = d,x,0,d; print(m);
==> d,x,
==> 0,d
list J = jacobson(m); // returns a list with 3 entries
print(J[2]); // a Jacobson Form D for m
==> xd2-d,0,
==> 0, 1
print(J[1]*m*J[3] - J[2]); // check that U*M*V = D
==> 0,0,
==> 0,0
/* now, let us do the same for the shift algebra */
ring r2 = 0,(x,s),Dp;
def R2 = nc_algebra(1,s); setring R2; // the 1st shift algebra
matrix m[2][2] = s,x,0,s; print(m); // matrix of the same for as above
==> s,x,
==> 0,s
list J = jacobson(m);
print(J[2]); // a Jacobson Form D, quite different from above
==> xs2+s2,0,
==> 0, x
print(J[1]*m*J[3] - J[2]); // check that U*M*V = D
==> 0,0,
==> 0,0
| See also:
divideUnits;
smith.
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