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From: Mark Adler on 27 May 2010 06:44
On 2010-05-26 04:10:18 -0700, Roland Franzius said:
> The second product is mathematically incorrect in the context of general
> matrix multiplication because a matrix product of 3x3 . 1x3 does not
> exist.

That vector does not have dimensions 1x3. Its dimensions are simply 3.
It is not a 2-dimensional matrix:

Dimensions[{1, 2, 3}]
{3}
Dimensions[{{1, 2, 3}}]
{1, 3}
Dimensions[{{1}, {2}, {3}}]
{3, 1}

Mathematica is being completely consistent on how those are treated for
an inner product, where simply the inner two dimensions must be equal
and are removed:

m = {{1, 2}, {3, 4}};
Dimensions[m]
{2, 2}
Dimensions[{1, 2}.m]
{2}
Dimensions[{{1, 2}}.m]
{1, 2}
Dimensions[m.{1, 2}]
{2}
Dimensions[m.{{1}, {2}}]
{2, 1}

Mathematica does complain about improper matrix products, when the
arguments are in fact matrices, e.g. 2x2 . 1x2:

m.{{1, 2}}
Dot::dotsh: Tensors {{1,2},{3,4}} and {{1,2}} have incompatible shapes.
{{1, 2}, {3, 4}}.{{1, 2}}

Mark

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