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
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