Eigenvalues of matrices of Laurent polynomials

asked 2015-08-14 16:26:16 +0100

fuglede
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updated 2015-10-16 19:18:09 +0100

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Say that I have a matrix whose entries are univariate Laurent polynomials over complex numbers. In that case, I can substitute the variable for non-zero complex numbers to obtain an ordinary complex matrix. My question is the following: what the correct way to obtain the eigenvalues of the resulting complex matrix? My first guess was to do something like the following which just throws a NotImplementedError:

sage: R = LaurentPolynomialRing(CC, 'x')
sage: x = R.gens()[0]
sage: m = matrix([[x]])
sage: m.subs({x: 2}).eigenvalues()

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answered 2015-08-14 20:22:36 +0100

nbruin
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One structural way of obtaining the answer you want is to first define the homomorphism from the laurent series ring to CC (or whatever ring you want); in your case the one obtained from substituting a particular complex value for x:

sage: s=Hom(R,CC)(2)
sage: s
Ring morphism:
  From: Univariate Laurent Polynomial Ring in x over Complex Field with 53 bits of precision
  To:   Complex Field with 53 bits of precision
  Defn: 1.00000000000000*x |--> 2.00000000000000

Making sure this gets applied to a matrix properly is now just (tedious) bookkeeping:

sage: mm=matrix(s.codomain(),m.nrows(),m.ncols(),[s(n) for n in m.list()] )
sage: mm.eigenvalues()
... UserWarning ...
[2.00000000000000]

Ideally, sage would know how to do the bookkeeping so that the definition of mm above can be written as

sage: mm=m.change_ring(s)
<doesn't work currently>

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Thanks for the answer. I'd upvote both of them if I had the karma.

fuglede ( 2015-08-18 16:56:22 +0100 )edit

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answered 2015-08-14 16:45:39 +0100

vdelecroix
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updated 2015-08-15 11:45:23 +0100

Hello,

The problem with your code is that the substitution does a too naive substitution

sage: R = LaurentPolynomialRing(CC, 'x')
sage: x = R.gens()[0]
sage: m = matrix([[x]])
sage: mm = m.subs({x: 2})
sage: mm.base_ring()
Univariate Laurent Polynomial Ring in x over Complex Field with 53 bits of precision

In order to get eigenvalues, you need to say explicitely that the matrix is now defined over complex numbers

sage: mm.change_ring(CC).eigenvalues()
... UserWarning ...
[2.00000000000000]

EDIT: as mentioned in the comment, the above solution does not work. This is because there is no way to initialize a complex number from a power series, even if it is constant

sage: a = R(CC(2,1))
sage: CC(a)
Traceback (most recent call last):
...
TypeError: unable to coerce to a ComplexNumber: <type 'sage.rings.power_series_poly.PowerSeries_poly'>

In the case the constant is real it works by some strange magic.

Vincent

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Thanks for the answer. That almost works. If I replace the '2' of the example by a non-real, though, I get a TypeError instead. Can anything be done about that?

fuglede ( 2015-08-14 17:04:18 +0100 )edit

I edited my answer...

vdelecroix ( 2015-08-15 11:45:57 +0100 )edit

I also opened a sage-devel thread to modify matrix.subs(). It would also be cool to allow constant power series to be converted to their base ring.

vdelecroix ( 2015-08-15 12:04:22 +0100 )edit

Nifty. Hope that one goes through. I've accepted the other answer as it does the job for now.

fuglede ( 2015-08-18 16:57:25 +0100 )edit

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