Can I get the transformation matrix for echelon form reduction over a finite field?

[Boyan Kostadinov]

I want to get the transformation matrix from the echelon form reduction over finite fields but I found in the SageMath documentation the following statement: 

The matrix library used for Z/pZ/p-matrices does not return the transformation matrix, so the transformation option is ignored

Link: http://doc.sagemath.org/html/en/reference/matrices/sage/matrix/matrix2.html#sage.matrix.matrix2.Matrix.echelon_form 

More specifically, the call:  M.echelon_form(transformation=True) does not return the transformation matrix T when M is over a finite field, only the echelon matrix E; T is such that we have T*M=E, so T is just the product of the elementary matrices, which perform the reducing row operations. 

Is there any workaround to get the transformation matrix T over a finite field? Can I get the transformation matrix T over ZZ and then reduce it over the finite field?

Any advice would be much appreciated.

[dim...@gmail.com]

On Fri, Oct 18, 2019 at 09:09:01PM -0700, Boyan Kostadinov wrote:
> I want to get the transformation matrix from the echelon form reduction
> over finite fields but I found in the SageMath documentation the following
> statement:
>

> *The matrix library used for Z/pZ/p-matrices does not return the
> transformation matrix, so the transformation option is ignored*
>
> Link:
> http://doc.sagemath.org/html/en/reference/matrices/sage/matrix/matrix2.html#sage.matrix.matrix2.Matrix.echelon_form
>
>
> More specifically, the call: *M.echelon_form(transformation=True**) *does

> not return the transformation matrix T when M is over a finite field, only
> the echelon matrix E; T is such that we have T*M=E, so T is just the
> product of the elementary matrices, which perform the reducing row
> operations.
>
> Is there any workaround to get the transformation matrix T over a finite
> field? Can I get the transformation matrix T over ZZ and then reduce it
> over the finite field?

Yes, this should work (with a loss in efficiency, which can be
substantial on large matrices). Anyway, this is how to do this.

sage: MS = MatrixSpace(GF(19),2,3)
....: C = MS.matrix([1,2,3,4,5,6])
....: E,T = map(lambda t: matrix(GF(19),t), matrix(ZZ,C).echelon_form(transformation=True))
....: T*C == E
....: E in MS
....:
True
True

---------------------------------------------------------------

HTH
Dmitrii

>
> Any advice would be much appreciated.
>

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[Boyan Kostadinov]

Thank you so much for the prompt and helpful reply Dmitrii! Much appreciated. 

Best,

Boyan

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