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

[Raoul]

Dear Tim,


> 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

Yes, I know this book very well. It would be indeed nice
if Axiom could do such things in an abstract fashion.


> 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'.

I think it would be possible, however I expect representation of
abstract matrix algebra to be much more complicated than the
vector algebra stuff. There are transposes, conjugates, inverses, 
traces, determinants and much more. One would need to have a way
to encode all these operation "unevaluated".


> 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.

Yes, of course this should play together with usual
(dense or sparse) matrices nicely. Doesn't make the task
easier though.

And do we want define abstract matrices with or without shape information?


> 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.

Maybe someone should indeed try. I don't think I find the
time to try this soon.


It occurred to me that component-free geometric algebra
might be easier to start with anyway.

There is a nice book on the topic:

  https://sites.google.com/site/grassmannalgebra/
  https://sites.google.com/site/grassmannalgebra/thegrassmannalgebrabook

which is accompanied by a Mathematica Package (that I never tried so far):

  https://sites.google.com/site/grassmannalgebra/thegrassmannalgebrapackage

The algebra in there is as far as I remember not done in an
abstract way, but with explicit bases.


-- Raoul