Not exactly, when one asks if two linear maps are equal it is not the same
thing to ask if they are similar, I mean equal as maps. In my example, if
you enter:
K=GF(2)
V=K^2
B=V.basis()
B1=[vector(K,[i,j]) for i,j in [(1,0),(1,1)]]
V1=V.subspace_with_basis(B1)
and you ask V==V1 the answer is True despite they are distinct objects in
Sage as their associated bases are not the same. In my opinion the option
is correct because as mathematical objects they are the same.
When one defines morphisms with domains V or V1, and same codomain, the
method keeps the matrix as comparison object. As you say computing the
Smith form (or Jordan form in case of endomorphisms) is expensive. I am not
sure if computing the images of the basis elements of either V or V1 is so
expensive. If it is so, it would be better to avoid an answer for the
equality, to avoid wrong answers.
This is the source of a._richcmp_??
def _richcmp_(self, other, op):
if not isinstance(other, MatrixMorphism) or op not in (op_EQ, op_NE): #
Generic comparison
return sage.categories.morphism.Morphism._richcmp_(self, other, op)
return richcmp(self.matrix(), other.matrix(), op)
File:
/usr/local/sage/local/lib/python3.8/site-packages/sage/modules/matrix_morphism.py
Type: method
Best, Enrique.
El jueves, 24 de diciembre de 2020 a las 0:41:10 UTC+1, Travis Scrimshaw
escribió:
> What you're asking for is checking if two matrices defining the linear
> transformations are similar, which involves computing the Smith normal
> forms of the matrices. This is computationally expensive. Instead, what we
> do is == becomes a weaker form of equality that is fast (equivalently has
> stronger conditions). This is something we do for graphs where we have a
> separate method is_isomorphic(). Therefore, I think that equal linear maps
> not being == is okay if they are given by different matrices.
>
> That being said, there is a definite issue with the false equality a == c,
> which because
>
> sage: a.parent()(c)
> Vector space morphism represented by the matrix:
> [1 0]
> [0 1]
> Domain: Vector space of dimension 2 over Finite Field of size 2
> Codomain: Vector space of dimension 2 over Finite Field of size 2
>
> which is not the correct map. This is a result of the fact the __call__
> does not check that the the (co)domains match and compose with the
> corresponding coercion maps.
>
> Best,
> Travis
>
>
> On Thursday, December 24, 2020 at 2:07:35 AM UTC+10 enriqu...@gmail.com
> wrote:
>
>> I have posted a bug in hom method for vector spaces in
>> https://groups.google.com/g/sage-support/c/1VDRFjZePCo/m/sFimwoV2BgAJ
>>
>> If one defines linear mappings using hom, distinct morphisms may be told
>> to be equal and viceversa
>>
>> Following a colleague advice (Miguel Marco) I looked at the method
>> _richcmp_ and it seems that it checks if the matrices of the morphisms are
>> equal. Since the same vector space can be defined in Sage with distinct
>> associated bases, this is the reason of the error.
>>
>> I guess that equality of domains (and maybe codomains) is the first
>> condition to be checked and then the equality of the images of self.
>>
>> Best, Enrique.
>>
>
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