#14117: Jordan normal form not computed for nilpotent matrix over rational
function
field / polynomials cannot be factored over rational function field
------------------------------------------------------------------+---------
Reporter: darij |
Owner: AlexGhitza
Type: defect |
Status: needs_review
Priority: major |
Milestone: sage-5.10
Component: algebra |
Resolution:
Keywords: polynomials, factorization, jordan normal form | Work
issues:
Report Upstream: N/A |
Reviewers:
Authors: Darij Grinberg | Merged
in:
Dependencies: |
Stopgaps:
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Changes (by {'newvalue': u'Darij Grinberg', 'oldvalue': ''}):
* cc: itolkov, sage-combinat (added)
* status: new => needs_review
* author: => Darij Grinberg
Old description:
> The manual for the jordan_form method on a matrix explicitly claims that
> the Jordan form is computed over arbitrary fields as long as the
> characteristic polynomial splits over there. This should particularly
> imply that the Jordan normal form of a nilpotent matrix is always
> computed. Unfortunately, this is not so:
>
> {{{
> sage: Qx = PolynomialRing(QQ, 'x11, x12, x13, x21, x22, x23, x31, x32,
> x33')
> sage: x11, x12, x13, x21, x22, x23, x31, x32, x33 = Qx.gens()
> sage: M = matrix(Qx, [[0, 0, x31], [0, 0, x21], [0, 0, 0]])
> sage: M**3
> [0 0 0]
> [0 0 0]
> [0 0 0]
> sage: N = M.base_extend(Qx.fraction_field())
> sage: N
> [ 0 0 x31]
> [ 0 0 x21]
> [ 0 0 0]
> sage: N.base_ring()
> Fraction Field of Multivariate Polynomial Ring in x11, x12, x13, x21,
> x22, x23, x31, x32, x33 over Rational Field
> sage: N.jordan_form()
> ---------------------------------------------------------------------------
> NotImplementedError Traceback (most recent call
> last)
>
> /home/darij/sage-5.6/<ipython console> in <module>()
>
> /home/darij/sage-5.6/local/lib/python2.7/site-
> packages/sage/matrix/matrix2.so in sage.matrix.matrix2.Matrix.jordan_form
> (sage/matrix/matrix2.c:43627)()
>
> /home/darij/sage-5.6/local/lib/python2.7/site-
> packages/sage/rings/polynomial/polynomial_element.so in
> sage.rings.polynomial.polynomial_element.Polynomial.roots
> (sage/rings/polynomial/polynomial_element.c:35063)()
>
> NotImplementedError: root finding for this polynomial not implemented
> }}}
>
> This seems to boil down to polynomial factorization over function fields
> not being implemented:
>
> {{{
> sage: N.characteristic_polynomial().factor()
> ---------------------------------------------------------------------------
> NotImplementedError Traceback (most recent call
> last)
>
> /home/darij/sage-5.6/<ipython console> in <module>()
>
> /home/darij/sage-5.6/local/lib/python2.7/site-
> packages/sage/rings/polynomial/polynomial_element.so in
> sage.rings.polynomial.polynomial_element.Polynomial.factor
> (sage/rings/polynomial/polynomial_element.c:24161)()
>
> NotImplementedError:
> }}}
>
> Unfortunately, I have no idea how to debug, let alone fix, things in C
> code, so I have nothing positive to contribute on this issue. Maybe a
> workaround is adding a new key "nilpotent" to the jordan_form method
> which, when set to True, skips the factorization of the characteristic
> polynomial and lets sage know that the matrix is nilpotent? In my
> personal experience, nilpotent matrices are the ones with the most
> interesting Jordan forms, and skipping the useless factorization would
> probably save quite some CPU cycles for them.
New description:
The manual for the jordan_form method on a matrix explicitly claims that
the Jordan form is computed over arbitrary fields as long as the
characteristic polynomial splits over there. This should particularly
imply that the Jordan normal form of a nilpotent matrix is always
computed. Unfortunately, this is not so:
{{{
sage: Qx = PolynomialRing(QQ, 'x11, x12, x13, x21, x22, x23, x31, x32,
x33')
sage: x11, x12, x13, x21, x22, x23, x31, x32, x33 = Qx.gens()
sage: M = matrix(Qx, [[0, 0, x31], [0, 0, x21], [0, 0, 0]])
sage: M**3
[0 0 0]
[0 0 0]
[0 0 0]
sage: N = M.base_extend(Qx.fraction_field())
sage: N
[ 0 0 x31]
[ 0 0 x21]
[ 0 0 0]
sage: N.base_ring()
Fraction Field of Multivariate Polynomial Ring in x11, x12, x13, x21, x22,
x23, x31, x32, x33 over Rational Field
sage: N.jordan_form()
---------------------------------------------------------------------------
NotImplementedError Traceback (most recent call
last)
/home/darij/sage-5.6/<ipython console> in <module>()
/home/darij/sage-5.6/local/lib/python2.7/site-
packages/sage/matrix/matrix2.so in sage.matrix.matrix2.Matrix.jordan_form
(sage/matrix/matrix2.c:43627)()
/home/darij/sage-5.6/local/lib/python2.7/site-
packages/sage/rings/polynomial/polynomial_element.so in
sage.rings.polynomial.polynomial_element.Polynomial.roots
(sage/rings/polynomial/polynomial_element.c:35063)()
NotImplementedError: root finding for this polynomial not implemented
}}}
This seems to boil down to polynomial factorization over function fields
not being implemented:
{{{
sage: N.characteristic_polynomial().factor()
---------------------------------------------------------------------------
NotImplementedError Traceback (most recent call
last)
/home/darij/sage-5.6/<ipython console> in <module>()
/home/darij/sage-5.6/local/lib/python2.7/site-
packages/sage/rings/polynomial/polynomial_element.so in
sage.rings.polynomial.polynomial_element.Polynomial.factor
(sage/rings/polynomial/polynomial_element.c:24161)()
NotImplementedError:
}}}
Unfortunately, I have no idea how to debug, let alone fix, things in C
code, so I have nothing positive to contribute on this issue. Maybe a
workaround is adding a new key "nilpotent" to the jordan_form method
which, when set to True, skips the factorization of the characteristic
polynomial and lets sage know that the matrix is nilpotent? In my personal
experience, nilpotent matrices are the ones with the most interesting
Jordan forms, and skipping the useless factorization would probably save
quite some CPU cycles for them.
'''Update:''' Here's a noobish patch, which adds an optional parameter to
the jordan_form method allowing pre-computed factorizations of the
characteristic polynomial. What do you guys think about it?
(This doesn't mean that we don't need a polynomial factorization algorithm
for multivariate polynomial rings over QQ; but I guess that's for a
different patch. We might want to do Jordan forms of, say, nilpotent
matrices over fields where factorization isn't even theoretically
possible.)
* Apply: [attachment:trac_14117-jordan_normal_form-v1.py]
--
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/14117#comment:1>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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