On Sat, Jun 6, 2009 at 2:30 PM, Robert Kern
On Sat, Jun 6, 2009 at 14:59, Alan G Isaac
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