On December 3, 2025 12:51:20 PM CST, Vincent Delecroix
<[email protected]> wrote:
>To my mind, it would be best to make eigenvalues actually have the
>same behavior as polynomials
>
>sage: QQ["x"]("x^2 + 1").roots() # only roots in the base ring
>[]
>sage: QQbar["x"]("x^2 + 1").roots()
>[(-1*I, 1), (1*I, 1)]
>
>Polynomials actually allow a nice syntax to search roots somewhere
>else with the extra argument "ring"
>
>sage: QQ["x"]("x^2 + 1").roots(QQbar)
>[(-1*I, 1), (1*I, 1)]
>sage: QQ["x"]("x^2 + 1").roots(ring=QQbar)
>[(-1*I, 1), (1*I, 1)]
+1 for adding "ring" option for eigenvalues/vectors
>
>Vincent
>
>On Wed, 3 Dec 2025 at 18:47, Nils Bruin <[email protected]> wrote:
>>
>> Your example hides a lesser consistency: the list of eigenvalues for a
>> matrix over QQ does seem to lie either in Q (if all the eigenvalues lie in
>> QQ) or in QQbar. Putting them in QQbar in all cases is of course the general
>> thing to do, but if you know your matrix has rational eigenvalues and you
>> just want to work with them (take their numerator and denominator perhaps?
>> reduce them mod p?) it could be quite annoying to find them in QQbar.
>> Arithmetic in QQbar is also going to be horribly slow in comparison.
>>
>> I can see why, with such an expensive general parent, there might be an
>> argument to put the arguments in a smaller parent when possible (for all of
>> them). There is a practical argument to be made here for why one might want
>> to deviate from the purity of "parent of return value determined by parent
>> of input".
>>
>> The documentation is formulated in a confusing way: It can be read as if for
>> matrices over QQ, only *separate* roots are returned: no multiplicity
>> information. I haven't tested this, but I'm pretty sure that's not what is
>> intended.
>>
>> The documentation also doesn't reflect the "parent optimization". I think
>> that's your main question. As described above, I think there's some
>> practical nuance to that...
>>
>> In actuality, due to the problem "eigenvalues in what field"? I'd probably
>> steer away from routines like "eigenvalues" in my own code, because I
>> wouldn't dare to trust the choice that's made. I probably instead would just
>> ask for the characteristic poly and determine its roots where I want (that's
>> almost never QQbar unless it's in an interactive situation), and then work
>> with that.
>>
>>
>> On Wednesday, 3 December 2025 at 01:52:39 UTC-8 [email protected] wrote:
>>>
>>> Do we really want that the parent of the result depends on the *value* of
>>> its input here, rather than on the parent of the input?
>>>
>>> Actually, strictly speaking the documentation promises something else:
>>>
>>> Return a sequence of the eigenvalues of a matrix, with
>>> multiplicity. If the eigenvalues are roots of polynomials in ``QQ``,
>>> then ``QQbar`` elements are returned that represent each separate
>>> root.
>>>
>>> Do I misunderstand this, or *is* the behaviour a bug? (If so, it is easy
>>> to fix.)
>>>
>>> I am guessing that harmonizing the hash of QQbar with the hash of QQ is
>>> expensive. Also, I don't think it would be the correct fix.
>>>
>>> Martin
>>> On Tuesday, 2 December 2025 at 23:20:30 UTC+1 Martin R wrote:
>>>>
>>>> sage: OZ = lambda P: (ones_matrix(P.cardinality()) - P.lequal_matrix())
>>>> sage: set(min(OZ(P).eigenvalues()) for n in range(1, 7) for P in posets(n))
>>>> {-1, -1, 0}
>>>> sage: [type(x) for x in _]
>>>> [<class 'sage.rings.rational.Rational'>,
>>>> <class 'sage.rings.rational.Rational'>,
>>>> <class 'sage.rings.qqbar.AlgebraicNumber'>]
>>>>
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>
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