Jordan normal form not computed for nilpotent matrix over rational function field / polynomials cannot be factored over rational function field

[Darij Grinberg]

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: 

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.)

Update 3: Better version.

• Apply: trac_14117-jordan_normal_form-dg-v4.patch​

[Darij Grinberg]

slightly improved

[Igor Tolkov]

Thanks for the CC. I'm somewhat new here, anything I should do?

[Darij Grinberg]

Hi -- I cc'ed you since I saw you working on the Jordan normal form today, so I'm wondering if you can provide some comments, maybe even a review...

[Luis Felipe Tabera Alonso]

With your example

sage: M.jordan_form(eigenvalues=[(0,3)])
[0 1|0]
[0 0|0]
[---+-]
[0 0|0]
sage: M.jordan_form(eigenvalues=[(1,3)])
[]

The code should not return the second but raise a type of error. Also, add to the doctest a test with your eigenvalues option with transformation=True to make sure it works as expected.

As a matter of style, avoid blank lines containing only blanckspaces (up to the indent level) in your code. I think this is not enforced though.

[Darij Grinberg]

Good point about the error raising! I'll do it later today, when I have time. I am going to add a check variable that defaults to True so that the checking is only done when it is True, though; otherwise the speed advantage of this might be significantly diminished. (Checking is best done by just comparing the characteristic polynomial to the product of (X-eigenvalue)^{(multiplicity) , or is there a better way?)}

I didn't know that blank lines should not be indented! This makes my life easier, since I actually thought they *should* be indented...

[Darij Grinberg]

version 2

[Darij Grinberg]

Oooops, I've just noticed that I gave the patch the extension .py rather than .patch. Anyway, here comes a new version, with a few typos fixed.

[Darij Grinberg]

version 3

[Darij Grinberg]

patchbot:

apply trac_14117-jordan_normal_form-dg-v3.patch

[Darij Grinberg]

Any idea why one of the patchbot machines reports PluginFailed??

[Travis Scrimshaw]

One very minor point, instead of Polyring(1), it's better to use Polyring.one() to take advantage of caching. Once you've made this change, you can set this to positive review on my behalf.

Best,
Travis

[Darij Grinberg]

reviewed version

[Darij Grinberg]

reviewed version

[Darij Grinberg]

Thank you!

For the patchbot:

apply trac_14117-jordan_normal_form-dg-v4.patch