On Sat, Jun 6, 2009 at 2:30 PM, Robert Kern <robert.kern@gmail.com> wrote:

On Sat, Jun 6, 2009 at 14:59, Alan G Isaac <aisaac@american.edu> wrote:
> On 6/6/2009 2:58 PM Charles R Harris apparently wrote:
>> How about the common expression
>> exp((v.t*A*v)/2)
>> do you expect a matrix exponential here?
>
>
> I take your point that there are conveniences
> to treating a 1 by 1 matrix as a scalar.
> Most matrix programming languages do this, I think.
> For sure GAUSS does. The result of x' * A * x
> is a "matrix" (it has one row and one column) but
> it functions like a scalar (and even more,
> since right multiplication by it is also allowed).
>
> While I think this is "wrong", especially in a
> language that readily distinguishes scalars
> and matrices, I recognize that many others have
> found the behavior useful. And I confess that
> when I talk about quadratic forms, I do treat
> x.T * A * x as if it were scalar.

The old idea of introducing RowVector and ColumnVector would help
here. If x were a ColumnVector and A a Matrix, then you can introduce
the following rules:

x.T is a RowVector
RowVector * ColumnVector is a scalar
RowVector * Matrix is a RowVector
Matrix * ColumnVector is a ColumnVector

Yes, that is another good solution. In tensor notation, RowVectors have signature r_i, ColumnVectors c^i, and matrices M^i_j. The '*' operator is then a contraction on adjacent indices, a result with no indices is a scalar, and the only problem that remains is the tensor product usually achieved by x*y.T. But making the exception that col * row is the tensor product producing a matrix would solve that and still raise an error for such things as col*row*row. Or we could simply require something like bivector(x,y)

Chuck