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matrix over symbolic ring to matrix over finite field I have a bunch of ideals defined symbolically, that I would like t

matrix over symbolic ring to matrix over finite field I have a bunch of ideals defined symbolically, that I would like t

Thank you very much. We seem to have found a similar solution, which does not obviously involve replacing variables; nam

Thank you very much. We seem to have found a similar solution, which does not obviously involve replacing variables; nam

@rburing my bad, thanks - added how the x's appear in the matrix. How would you skip the symbolic ring if your matrices

matrix over symbolic ring to matrix over finite field I have a bunch of ideals defined symbolically, that I would like t

Edited. No self-contained code. But enough detail I think to render the issue intelligible.

matrix over symbolic ring to matrix over finite field I have a bunch of ideals defined symbolically, that I would like t

matrix over symbolic ring to matrix over finite field I have a bunch of ideals defined symbolically, that I would like t

A very belated thank you!

I have the following expression:

f = u^2*v*x + 3*u*v^2*x + 2*v^3*x + u^2*x^2 + 5*u*v*x^2 + 5*v^2*x^2 + 2*u*x^3 + 4*v*x^3 + x^4 + 2*u^2*v*y + 6*u*v^2*y + 4*v^3*y + 2*u^2*x*y + 12*u*v*x*y + 12*v^2*x*y + 6*u*x^2*y + 12*v*x^2*y + 4*x^3*y + u^2*y^2 + 6*u*v*y^2 + 6*v^2*y^2 + 6*u*x*y^2 + 12*v*x*y^2 + 5*x^2*y^2 + 2*u*y^3 + 4*v*y^3 + 2*x*y^3 + u^2*v*z + 3*u*v^2*z + 2*v^3*z + u^2*x*z + 6*u*v*x*z + 6*v^2*x*z + 3*u*x^2*z + 6*v*x^2*z + 2*x^3*z + u^2*y*z + 6*u*v*y*z + 6*v^2*y*z + 6*u*x*y*z + 12*v*x*y*z + 5*x^2*y*z + 3*u*y^2*z + 6*v*y^2*z + 3*x*y^2*z + u*v*z^2 + v^2*z^2 + u*x*z^2 + 2*v*x*z^2 + x^2*z^2 + u*y*z^2 + 2*v*y*z^2 + x*y*z^2

and I would like to simplify it by gathering and factoring partial sums

u, u+v,...,u+v+x+y+z
v, v+x,...,v+x+y+z
x, x+y,x+y+z,
y,y+z
z

(I have reason to expect that the expression will look quite a bit nicer if I do this, introducing notation for the sums of course, like h_{ij} = the sum of variables i up to j)

My first lame try at this was to ask sage to rational_simplify() quotients f/(u+v+x+y+z),...,f/(u+v), and so on, hoping to see more than just one big fraction. It didn't work.

Any insights/suggestions/corrections would be much appreciated!

Thanks,

Thanks very much!

Yes, it might. How do I do it, exactly?

I would work in the ring $R/I$, where $R$ is the polynomial ring generated by matrix coefficients, and $I$ is the ideal of the Jordan form. For instance, if I have a 5 by 5 matrix $A$ of type $(3,2)$ then

• rank A = 3 would be encoded by adding all 4 by 4 minors of A to I
• rank A^2 = 1 would be encoded by adding all 2 by 2 minors of A^2 to I
• and finally rank A^3 = 0 would be encoded by adding coefficients of A^3 to I

In general, the constraints may be more complicated than setting some entries to zero. Either way, I want Sage to figure out those constraints for me, and solve something like $g A g^{-1} = J$ for a $g$, given variable $A$ and fixed $J$.

If I try A.jordan_form() in a quotient ring, I get a NotImplementedError :(

Sure. Say I have an arbitrary uppertriangular matrix of Jordan type $(2,1)$.

$$ A = \begin{pmatrix} 0 & A_{0} & A_{1} \\ 0 & 0 & A_{2} \\ 0 & 0 & 0 \end{pmatrix} $$

By default J_d, T_d = A.jordan_form(transformation=True) gives

$$ J_d = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix} \qquad T_d = \begin{pmatrix}A_{0} A_{2} & A_{1} & 0 \\ 0 & A_{2} & 0 \\ 0 & 0 & 1\end{pmatrix} $$

However, I expect $$J = \begin{pmatrix}0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{pmatrix}$$

In this case, I know that $A_2$ must be zero for $A$ to have Jordan type $(2,1)$ so I let $A' = A\big|_{A_2 = 0}$ and do J, T = A'.jordan_form(transformation=True) getting the desired

$$T = \begin{pmatrix}A_{0} & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & -\frac{A_{0}}{A_{1}}\end{pmatrix}$$

related: question/10488/why-does-jordan_form-not-work-over-inexact-rings/

I have, say, a nilpotent uppertriangular matrix $A$, with variable entries, and of a given Jordan type, $J$ (block type $\lambda$). I would like to know the transformation matrix $g$ such that $g A g^{-1} = J$.

How do I tell sage my input has a given Jordan type?

My idea is to compute A.jordan_form(transformation=True) in the quotient ring of the variable entries, given $\lambda$. This requires my figuring out the relations imposed by the given Jordan type, itself not an easy task.

I would like to know if what I want is already implemented somewhere in Sage, or if there is a better way to implement it than what I propose.