I think a lot of the matrix code is assumed to work over commutative rings,
but there is no assumption on them being fields. The MixedFormAlgebra is
not commutative, correct? If so, I think the best thing is to implement
what you need (i.e., the determinant) as a separate function for now.
Otherwise, you should mark it in Algebras(R).Commutative() and you should
be able to just use the generic matrix class and its determinant in Sage.
sage: R.<x,y,z> = QQ[]
sage: Q = R.quotient([x^2*y-x,y^2-z,z^2-2*x*y-x-y-z])
sage: M = matrix([[Q(R.random_element()) for _ in range(3)] for _ in range(3
)])
sage: M
[ -35/2*ybar*zbar + 1/2*zbar^2 - 1/2*xbar - 1/2*ybar + 1/2*zbar
xbar*zbar + 3*zbar^2 - 1/3*ybar + 2/3*zbar -13*xbar^2 - 3*xbar*zbar
- 1/2*ybar*zbar - 5*zbar^2 - 1/2*xbar]
[ 1/2*zbar^2 + 1/2*xbar - 3/2*ybar + 9/2*zbar
8*xbar*zbar + 2*zbar^2 - 1/38
2*xbar^2 - zbar^2 + ybar]
[ -1/6*ybar*zbar + 43/41*zbar^2 - xbar - ybar - zbar + 5/2
83/414*zbar^2 + 1/46*xbar + 1/46*ybar + 1/46*zbar -27/2*zbar^
2 + 191/14*xbar + 13*ybar + 25/2*zbar]
sage: M.det()
8984729371/2006704*zbar^3 + 8060049989/2257542*xbar^2 + 14983126743/2006704*
xbar*zbar - 78071057467/18060336*ybar*zbar + 2763514553/785232*zbar^2 +
648366185/125419*xbar - 23760880219/3010056*ybar - 573699071/73416*zbar
sage: type(M)
<type 'sage.matrix.matrix_generic_dense.Matrix_generic_dense'>
Best,
Travis
On Wednesday, May 15, 2019 at 3:34:51 AM UTC+10, Michael Jung wrote:
>
> Unfortunately, I noticed that the algorithm of computing the determinant
> of MixedForm elements doesn't work for matrices of dimension higher than 3
> since MixedFormAlgebra isin general not a field.
>
> So, is it useful to create a new matrix class for this kind of purpose? If
> so, how is the matrix implementation structured in sagemath? I couldn't
> find any helpful documentation.
>
> And furthermore, what would be an appropriate efficient algorithm to
> compute the determinant? Is it possible to use parallel processing, and if
> so, how?
>
> Best regards
> Michael
>
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