Add the outer product of two vectors

[rbeezer]

Title pretty much says it all. Students frequently can't tell an inner product from an outer product (probably not too careful about where a transpose is). Maybe this will help.

Depends on #10541

Apply trac_10545-vector-outer-product.patch, trac_10545-vector-outer-product-doc.patch

[rbeezer]

For anybody trying this out, it really only needs the row/column patch on #10541. You will get one documentation error, but that is fixed on the other half of #10541.

Patchbot:

Depends on #10541

[nbruin]

I didn't know that is sometimes used as a definition of "outer product", but Wikipedia backs you up on it.

In other languages, "exterior product" translates to the same word as "outer product", so a -1 from me for having this definition of outer product. I think this term will be a source of confusion.

If someone needs the tensor product, it is easy enough to get via the one-liner

lambda v,w: matrix(len(v),len(w),[a*b for a in v for b in w])

(given Sage's preference for row vectors versus the preference of most LA texts for column vectors, I expect that relating inner/outer products to a question of where to put the transpose is only going to cause *more* confusion in students, by the way)

[rbeezer]

Hi Nils,

Thanks for the comments. Since vectors are neither rows nor columns in Sage, any notion of a transpose is irrelevant. The outer product just is what it is. I could delete the mention of the transpose in the mathematical description in the docstring.

"Someone" are 19-year-old students, who shouldn't need to understand a lambda function to learn linear algebra.

Sage can work well with a column-oriented approach - it just needs a few things. Check out:

​http://wiki.sagemath.org/devel/LatexToWorksheet

​http://linear.ups.edu

Rob

Replying to nbruin:

I didn't know that is sometimes used as a definition of "outer product", but Wikipedia backs you up on it.

In other languages, "exterior product" translates to the same word as "outer product", so a -1 from me for having this definition of outer product. I think this term will be a source of confusion.

If someone needs the tensor product, it is easy enough to get via the one-liner

lambda v,w: matrix(len(v),len(w),[a*b for a in v for b in w])

(given Sage's preference for row vectors versus the preference of most LA texts for column vectors, I expect that relating inner/outer products to a question of where to put the transpose is only going to cause *more* confusion in students, by the way)

[nbruin]

"Someone" are 19-year-old students, who shouldn't need to understand a lambda function to learn linear algebra.

Slightly off-topic: If you don't want lambda and v,w are the vectors you want to take the kronecker product of:

matrix([[a*b for b in w] for a in v])

(to avoid depending on the order in which python iterates through a multiple "for")

On-topic: In Dutch "uitprodukt" and in German "auseres produkt" is "exterior product" (as in "wedge product" or "cross product"). These terms literally translate to "outer product". I don't know about french. That's why I think "outer product" is better avoided, because it likely causes confusion.

(truly off-topic: Do you think you are doing 19-year olds a service by teaching them kronecker products of column x row vectors? Cross products of 3-dim vectors are probably much more useful for them and having to juggle 3 (instead of 2) ways of taking products of vectors is probably more confusing for them)

[rbeezer]

NumPy calls it the outer product.

​http://docs.scipy.org/doc/numpy/reference/generated/numpy.outer.html#numpy.outer

[flawrence]

This definition of outer product was what I was taught, and it seems to be the standard terminology.

The code works and passes long doctests, and is well documented. Positive review.

[jdemeyer]

The ticket number in the commit message is wrong.

[rbeezer]

Jeroen - sorry for the trouble. Patch replaced with a fixed one.

[jdemeyer]

During a trial merge of sage-4.7.alpha3, I get the following doctest error:

sage -t  -force_lib devel/sage/sage/modules/free_module_element.pyx
**********************************************************************
File "/mnt/usb1/scratch/jdemeyer/merger/sage-4.7.alpha3/devel/sage-main/sage/modules/free_module_element.pyx", line 2292:
    sage: z = w.outer_product(v)
Expected:
    Traceback (most recent call last):
    ...
    TypeError: unsupported operand parent(s) for '*': 'Full MatrixSpace of 2 by 1 dense matrices over Rational Field' and 'Full MatrixSpac
e of 1 by 4 dense matrices over Finite Field of size 7'
Got:
    Traceback (most recent call last):
      File "/mnt/usb1/scratch/jdemeyer/merger/sage-4.7.alpha3/local/bin/ncadoctest.py", line 1231, in run_one_test
        self.run_one_example(test, example, filename, compileflags)
      File "/mnt/usb1/scratch/jdemeyer/merger/sage-4.7.alpha3/local/bin/sagedoctest.py", line 38, in run_one_example
        OrigDocTestRunner.run_one_example(self, test, example, filename, compileflags)
      File "/mnt/usb1/scratch/jdemeyer/merger/sage-4.7.alpha3/local/bin/ncadoctest.py", line 1172, in run_one_example
        compileflags, 1) in test.globs
      File "<doctest __main__.example_57[27]>", line 1, in <module>
        z = w.outer_product(v)###line 2292:
    sage: z = w.outer_product(v)
      File "free_module_element.pyx", line 2307, in sage.modules.free_module_element.FreeModuleElement.outer_product (sage/modules/free_mo
dule_element.c:12527)
        return self.column()*right.row()
      File "element.pyx", line 2282, in sage.structure.element.Matrix.__mul__ (sage/structure/element.c:15874)
      File "coerce.pyx", line 709, in sage.structure.coerce.CoercionModel_cache_maps.bin_op (sage/structure/coerce.c:6368)
      File "action.pyx", line 139, in sage.matrix.action.MatrixMatrixAction._call_ (sage/matrix/action.c:2742)
      File "matrix_rational_dense.pyx", line 1331, in sage.matrix.matrix_rational_dense.Matrix_rational_dense.change_ring (sage/matrix/mat
rix_rational_dense.c:13867)
      File "element.pyx", line 1551, in sage.structure.element.RingElement.__div__ (sage/structure/element.c:12720)
      File "coerce.pyx", line 713, in sage.structure.coerce.CoercionModel_cache_maps.bin_op (sage/structure/coerce.c:6426)
      File "element.pyx", line 1549, in sage.structure.element.RingElement.__div__ (sage/structure/element.c:12694)
      File "integer_mod.pyx", line 2223, in sage.rings.finite_rings.integer_mod.IntegerMod_int._div_ (sage/rings/finite_rings/integer_mod.
c:19384)
    ZeroDivisionError: Inverse does not exist.
**********************************************************************

[rbeezer]

Thanks, Jeroen. Coercion/error-trapping got changed somehow, and I've asked about it on sage-devel. New patch just adjusts the doctest to a pair of rings that behave better and make the same point, and this is really a matrix multiplication error message anyway.

I'll see if I can get a review here at Sage Days 29. ;-)

Rob

[jhpalmieri]

Should this be compared with the "tensor_product" method of matrices? Should "outer_product" have "tensor_product" as a synonym? (According to wikipedia, it should.) For reasons I don't understand, the tensor product of matrices comes equipped with subdivisions, but other than that,

sage: v.outer_product(w)

and

sage: m1 = matrix(v).transpose()
sage: m2 = matrix(w)
sage: m1.tensor_product(m2)

should produce the same thing.

[rbeezer]

Replacement "doc" patch is still almost all documentation, but now includes a "tensor_product" synonym as suggested.

New doctest compares it with the matrix version of a tensor product.

[rbeezer]

Latest addition passes all long tests, so is ready for review.