Re: [Axiom-mail] A question about Axiom capabilities

[Mike Valenzuela]

I just wanted to thank you all for taking this task seriously. I've been looking for some CAS that provided significant symbolic matrix manipulations, including matrix calculus. I found other tools that work with tensors and provide tensor calculus (Maxima CAS has 3 tensor packages), but second order tensors and matrices are not quite the same thing. Furthermore, the packages were so general that they lacked specific rules (derivatives of determinants, inverses, etc.) which worked fine on matrices.

So once again, just saying thanks.

On Thu, Feb 28, 2013 at 7:46 PM, u1204 <address@hidden> wrote:

Raoul,

If you look at the matrixcookbook that Mike mentioned, the first 10
equations are:

   (A*B)^-1      = B^-1 * A^-1
   (A*B*C...)^-1 = ...C^-1 * B^-1 * A^-1
   (A^T)^-1      = (A^-1)^T
   (A+B)^T       = A^T + B^T
   (A*B)^T       = B^T * A^T
   (A*B*C...)^T  = C^T * B^T *A^T
   (A^H)^-1      = (A^-1)^H
   (A+B)^H       = A^H + B^H
   (A*B)^H       = B^H * A^H
   (A*B*C...)^H  = ...C^H * B^H * A^H

It seems like we could create a SymbolicMatrix algebra that could
perform these manipulations with uninterpreted matrix symbols
A, B, C with a special recognized symbol 'T'.

These matrices could have actual values which, for certain operations
are ignored, so that

       (A*B)^-1      = B^-1 * A^-1

but for other operations would be evaluated as in:

       eval(B^-1 * A^-1)

giving the actual matrix result shown element by element.

An additional enhancement would be to make a SymbolicMatrixCategory
so that there could be specific domains such as

   GeneralSymbolicMatrix,
   SymmetricSymbolicMatrix,
   UpperTriangularSymbolicMatrix,
   DiagonalSymbolicMatrix,

etc which could exploit certain matrix-level properties at the
symbolic level.

This isn't exactly what Mike was originally asking but I think
that Axiom ought to be able to symbolically compute the equations
in the handbook.

Tim Daly