Can Maxima be used to simplify symbolic matrix algebra statements?

Stavros,

Thanks for taking the time to help. I'll try out your package and give 
feedback.

Rudolph

Stavros Macrakis wrote:

> Here are some notes on simplifying non-commutative operations in Maxima
> along with a small package I wrote which may be helpful.  For example,
> it simplifies some of the expressions in your original email, and can
> also prove the Sherman-Morrison-Woodbury matrix inversion identity with
> no manual intervention.
> 
> Maxima has limited built-in capabilities for simplifying non-commutative
> operators.  To use them effectively, it is important to explicitly
> declare scalar variables; in most cases, undeclared variables are
> treated as non-scalar, but it is best to declare them explicitly, too.
> It is also useful to check the various options starting DOT; the
> defaults are all reasonable, except that I usually also turn on
> Dotscrules.
> 
> There are share packages which may be relevant (affine and various
> tensor), but I don't know anything about them.
> 
> Even for monomials (expressions involving only "*", ".", and "^^"),
> Maxima has some limitations/bugs.  For example, though it does simplify
> a.b.(a.b)^^-1 to 1, it does not do so for a.b.(c.a.b)^^-1 (instead, use
> my workaround "dotexpand" function in the dotsimp package).  Even more
> blatantly: a^^-1 . (a.b)^^2 or even (a^^-1 . a . b . a . b) do not
> currently simplify to b.a.b (my bug report # 629716).  There are
> workarounds for that, too -- dotexpand(dotexpand(xxx^^-1)^^-1) -- but
> that is ridiculous.
> 
> For polynomials, the main value of Maxima is not the automatic
> simplifications (which are not very powerful), but rather the
> possibility of manipulating complicated expressions without error.
> 
> I have a very simple routine for some trivial cases of non-commutative
> factoring which I call simpledotfactor (attached).  It turns out that
> these trivial cases are enough to, for example, simplify your example:
> 
> Matlab: A*(B + inv(A)) * inv(A*B + I)
> Maxima: A.(B + A^^-1)  . (A.B + 1)^^-1
> 
> and for that matter to prove the Sherman-Morrison-Woodbury matrix
> inversion identity.  But it knows nothing about the transpose operator,
> and can't help for your other problem.
> 
> My dotsimp package requires the union function from the new set package,
> available from
> http://www.unk.edu/acad/math/people/willisb/nset-1.0.tar.gz (presumably
> at some point nset will be added to the share directories in CVS).
> 
> I hope you find this helpful -- I'm always interested in comments, bug
> reports, and suggestions.
> 
>       -s
> 
> Stavros Macrakis
> 
> 


-- 
Rudolph van der Merwe  |  http://bme.ogi.edu/~rvdmerwe