Eigenvalues of matrix with entries in polynomial ring

Anne Schilling
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slelievre
17789 ●22 ●163 ●352 http://carva.org/samue...

Hi!

I just wrote some code on the sage-combinat queue which computes a matrix with entries in a polynomial ring R = PolynomialRing(QQ, 'x', n)

sage: P = Poset(([1,2,3,4], [[1,3],[1,4],[2,3]]), linear_extension = True)
sage: L = P.linear_extensions()
sage: M = L.markov_chain_transition_matrix(labeling = 'source')
sage: M
[-x0 - x1 - x2            x3       x0 + x3             0             0]
[      x1 + x2 -x0 - x1 - x3             0            x1             0]
[            0            x1      -x0 - x3             0            x1]
[            0            x0             0 -x0 - x1 - x2       x0 + x3]
[           x0             0             0       x0 + x2 -x0 - x1 - x3]
sage: M.eigenvalues()
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)

/Applications/sage-5.0.beta7/devel/sage-combinat/sage/combinat/posets/<ipython console> in <module>()

/Applications/sage-5.0.beta7/local/lib/python2.7/site-packages/sage/matrix/matrix2.so in sage.matrix.matrix2.Matrix.eigenvalues (sage/matrix/matrix2.c:26415)()

/Applications/sage-5.0.beta7/local/lib/python2.7/site-packages/sage/matrix/matrix2.so in sage.matrix.matrix2.Matrix.fcp (sage/matrix/matrix2.c:11089)()

/Applications/sage-5.0.beta7/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:22655)()

NotImplementedError:

Is it possible to compute this some other way or is this just not yet implemented (which would surprise me!).

Thanks,

Anne

Edit (originally posted as an answer by the original poster of the question)

Here is an easier example with the question:

sage: R = PolynomialRing(QQ, 'x', 2)
sage: x = R.gens()
sage: M = matrix([[x[0],x[1]],[x[1],x[0]]])
sage: M.eigenvalues()
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)

/Applications/sage-5.0.beta7/devel/sage-combinat/sage/combinat/posets/<ipython console> in <module>()

/Applications/sage-5.0.beta7/local/lib/python2.7/site-packages/sage/matrix   /matrix2.so in sage.matrix.matrix2.Matrix.eigenvalues (sage/matrix/matrix2.c:26415)()

/Applications/sage-5.0.beta7/local/lib/python2.7/site-packages/sage/matrix/matrix2.so in sage.matrix.matrix2.Matrix.fcp (sage/matrix/matrix2.c:11089)()

/Applications/sage-5.0.beta7/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:22655)()

NotImplementedError:

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kcrisman ( 13 years ago )

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answered 13 years ago

DSM

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updated 6 years ago

nbruin
4153 ●3 ●45 ●91

Could you change the ring to SR as a workaround?

sage: R = PolynomialRing(QQ, 'x', 2)
sage: x = R.gens()
sage: M = matrix([[x[0],x[1]],[x[1],x[0]]])
sage: M.eigenvalues()
---------------------------------------------------------------------------
NotImplementedError                       Traceback (most recent call last)
[...]
sage: M.change_ring(SR)
[x0 x1]
[x1 x0]
sage: M.change_ring(SR).eigenvalues()
[x0 - x1, x0 + x1]

EDIT: You might want to use K=R.fraction_field() and then do something like M.characteristic_polynomial().change_ring(K).roots(). It doesn't get you the eigenvalues that aren't rational over R, but the answer you get back in SR will not be very useful either.

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Thank you! That works! Am I allowed to use this in sage source code?

Anne Schilling ( 13 years ago )

If by that you mean for stuff getting committed to mainline, I'm not sure. Seems a little hacky, and it feels like there should be a way to do it while staying purely in some polynomial ring.

DSM ( 13 years ago )

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answered 6 years ago

nbruin
4153 ●3 ●45 ●91

Root finding of polynomials over function fields seems to be implemented, so this should work:

sage: M.change_ring(R.fraction_field()).characteristic_polynomial().roots()
[(x0 + x1, 1), (x0 - x1, 1)]

It has the advantage that it should also work in positive characteristic (where SR would not), and hopefully good use is made of specifics of the coefficient domain to provide efficient arithmetic.

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