Hi
Id like to write a package that can do "pure"/"abstract" matrix
expressions such as in the following examples
To keep things simple, in the examples below, let
A, B, C... be matrices; k be a scalar
"*" be matrix or scalar multiplication, "^-1" and "^T" and "^n" be
matrix inverse, transpose and power
sage: (A * B)^2
A * B * A * B
sage: (A * B)^-1
B^-1 * A^-1
sage: assume(A = B^-1)
sage: A * B
I # this is the identity matrix
sage: A * (B+ C)
A * B + A * C
sage: (A * B + A * C).factor()
A * (B+ C)
sage: k * (B+ C)
k * B + k * C
# Note: Assume I probably did a few things first, like
sage: var('k,N'); A = abstractmatrix(N, N); B = abstractmatrix(N, N);
C = abstractmatrix(N, N)
# this is so sage knows what kind of objects "*" and "+" are acting on
and so I cant add/multiply nonconforming matrices
This isnt a full specification - there are many issues not addressed
and note we are considering "pure matrix" expressions only (i.e. no
matrix components). Also the matrix size "N" is arbitrary - doesnt
ever need to be assigned a value. I can think of two ways of doing
this and just asking for peoples gut feeling which direction I should
go in to implement this within Sage.
Possibility 1) We consider this a noncommutative algebra and look at
GAP (are there other possiblilities?)
Possibility 2) Consider these matrix expressions as symbol expressions
and use pattern matching with rewrite rules to transform expressions
such as those shown above (python has regular expression capabilities
and I expect something can be done using maxima or pynac - is there
anything else? Lisp maybe?)
I expect this will be wanted by some people. Some guidance will
hopefully realize it sooner.
BTW Its ok if the solution involves python, C++ or Lisp
Thanks
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