[Numpy-discussion] Further matrix oddities: no inner product

[Charles R Harris]

In mathematics, and inner product is a sesquilinear form on pairs of
vectors, so at the least it should return a scalar. In numpy inner is a sum
over the last indices. OK, so we have

In [10]: inner(ones(2),ones(2))
Out[10]: 2.0

This doesn't work as an inner product for column vectors, which would be the
usual textbook convention, but that's alright, it's not a 'real' inner
product. But what happens when matrices are involved?

In [11]: inner(mat(ones(2)),mat(ones(2)))
Out[11]: array([[ 2.]])

Hmm, we get an array, not a scalar. Maybe we can cheat

In [12]: mat(ones(2))*mat(ones(2)).T
Out[12]: matrix([[ 2.]])

What about vdot (conjugate of the mathematical convention, i.e., the Dirac
convention)

In [17]: vdot(mat(ones(2)),mat(ones(2)))
---------------------------------------------------------------------------
exceptions.ValueError                                Traceback (most recent
call last)

/home/charris/<ipython console>

ValueError: vectors have different lengths

In [18]: vdot(mat(ones(2)),mat(ones(2)).T)
---------------------------------------------------------------------------
exceptions.ValueError                                Traceback (most recent
call last)

/home/charris/<ipython console>

ValueError: vectors have different lengths


Nope, vdot doesn't work for row and column vectors. So there is *no* builtin
inner product that works for matrices. I wonder if we should have one, and
if so, what it should be called. I think that vdot should probably be
modified to do the job.  There is also the question of whether or not v.T *
v should be a scalar when v is a column vector. I believe that construction
is commonly used in matrix algebra as an alias for the inner product,
although strictly speaking it uses the mapping between a vector space and
its dual that the inner product provides.

Chuck
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