fuglede's profile - activity

Thanks for the answer. I don't think it entirely answers my question. In my case, there is a way to obtain a matrix over the desired ring as one could try to coerce all entries (even though apply_map doesn't work for whatever reason). Clearly though, that's a very inelegant way of doing it, and so I was looking for the "correct" way to do it.

This question is rather similar in spirit to my previous question in that it's about figuring out how to "correctly" manipulate matrices of polynomials, ensuring that everything stays in the right rings.

In this case, I have a matrix, all of whose elements are of the form Univariate Laurent Polynomial Ring in t over Integer Ring. The matrix has an inverse over this ring, yet sage insists that the entries of the inverse belong to Fraction Field of Univariate Polynomial Ring in t over Integer Ring. Here's a minimal example:

sage: R.<t> = LaurentPolynomialRing(ZZ)
sage: f = t^2
sage: mat = matrix([[f]])
sage: mat[0,0].parent()
Univariate Laurent Polynomial Ring in t over Integer Ring
sage: mat.inverse()
[1/t^2]
sage: mat.inverse()[0,0].parent()
Fraction Field of Univariate Polynomial Ring in t over Integer Ring

Desired output:

sage: mat.inverse()
[t^-2]
sage: mat.inverse()[0,0].parent()
Univariate Laurent Polynomial Ring in t over Integer Ring

What would have been the correct way to obtain what I want?

Using apply_map to force each entry back into the right ring does not immediately do the job:

sage: R(1/f)
t^-2
sage: mat.inverse().apply_map(lambda f: R(f))
...
TypeError: denominator must be a unit
sage: mat.inverse().apply_map(R)
...
TypeError: denominator must be a unit

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

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

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?

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()