Smith Form Decomposition

[Florent Bulkilol]

Hi,

I need to perform a Smith decomposition on (a priori not square) matrices to find a certain change of basis related to the Smith invariants. 

The Smith normal form is already implemented in SymPy in `sympy.matrices.normalforms`, but it returns only the Smith normal form, not its decomposition itself.

In general, the Smith decomposition of a matrix A is defined in terms of three matrices V, D, W so that

A = V * D * W

where V,W are both square, invertible, and integer-valued matrices. Is there any way of obtaining those matrices without reimplementing the algorithm myself? I assume that these matrices are computed at least implicitely, but I could not find a way of returning them.

thanks!

[Chris Smith]

As far as I can tell, these matrices are not computed implicitly. You would have to copy the appropriate actions onto an augmented identity matrix to see what has happened (https://www.youtube.com/watch?v=UhyzLfiO4Ow).

/c

[Oscar Benjamin]

It would be good to have this in SymPy but unfortunately it is not
implemented yet.

It is also not available in python-flint or Flint either.

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[Sangyub Lee]

You can check Add full option to smith_normal_form by isuruf · Pull Request #17451 · sympy/sympy (github.com).
Although the PR is based on outdated branch, you may have easier time following the algorithms based on that

[Chris Smith]

I left a modified version in the comment here,  Add full option to smith_normal_form by isuruf · Pull Request #17451 · sympy/sympy (github.com) , that will compute the decomposition.

[Oscar Benjamin]

Just following up on this. The Smith normal form with the transform
matrices is now available in SymPy's master branch (thanks to Isuru
Fernando):

https://github.com/sympy/sympy/pull/17451

To test it you can install sympy from master like:

pip install git+https://github.com/sympy/sympy.git@master

It returns the inverses of the matrices V and W that you were expecting:

In [4]: from sympy.matrices.normalforms import smith_normal_decomp

In [5]: A = randMatrix(3, 4)

In [6]: D, v, w = smith_normal_decomp(A)

In [7]: D == v*A*w
Out[7]: True

In [9]: V, W = v.inv(), w.inv()

In [10]: A == V*D*W
Out[10]: True

--
Oscar