4
$\begingroup$

I work with traces of long matrix products and I want mathematica to employ the cyclicity when simplifying.

I use this redefinition of the non-commutative multiplication:

Unprotect[NonCommutativeMultiply];
Clear[NonCommutativeMultiply]

(*Factor out numerics-- could generalize to some ScalarQ*)
nc : NonCommutativeMultiply[a__] /; MemberQ[{a}, _?NumericQ] := 
 NCMFactorNumericQ[NCM[a]] /. NCM -> NonCommutativeMultiply

(*Simplify Powers*)    
b___ ** a_^n_. ** a_^m_. ** c___ := 
 NCM[b, a^(n + m), c] /. NCM -> NonCommutativeMultiply

(*Minus*)
NonCommutativeMultiply[a___, b_Times, c___] := 
 NCMFactorNumericQ[NCM[a, Sequence @@ b, c]] /. 
  NCM -> NonCommutativeMultiply
Protect[NonCommutativeMultiply];

Unprotect[NCM];
Clear[NCM]
NCMFactorNumericQ[nc_NCM] := 
 With[{pos = Position[nc, _?NumericQ, 1]}, 
  Times @@ Extract[nc, pos] Delete[nc, pos]]
NCM[a_] := a
NCM[] := 1
Protect[NCM];

which was given as an answer to this question and is very helpful, but I want it to be able to cancel A**B**C with -B**C**A (but only if these matrices stand alone, i.e. I don't want it to cancel A**B**C**D with -B**C**A**D). Any help would be greatly appreciated.

$\endgroup$
2
  • $\begingroup$ Beware! By default, ** is associative, but one implication of the last statement is that your multiplication is not. Thus, even an expression like A**B**C is not well-defined, because it could mean either (A**B)**C or A**(B**C). You probably should not be thinking of your manipulations with traces as resulting from a binary operation at all. $\endgroup$
    – whuber
    Dec 13, 2012 at 15:25
  • 1
    $\begingroup$ Good point, @whuber . I was thinking something along the lines of letting it check if what is left and right is a one (then it can cycle) or something else (then it can't). On the other hand, I don't really need it to be encoded in the multiplication itself. I'd be perfectly happy with a newly defined trace that I can apply on my expressions and that knows that cyclic permutations are identical. $\endgroup$
    – Sebastik
    Dec 13, 2012 at 16:41

1 Answer 1

1
$\begingroup$

NCAlgebra supports a trace type operator. It will automatically reduce

tr[a ** b ** c - b ** c ** a]

to

0

and

tr[a ** b ** c ** d - b ** c ** a ** d]

tr[a ** b ** c ** d] - tr[a ** d ** b ** c]

It achieves that by automatically canonizing the sorting of the arguments or tr.

P.S.: We are in the process of updating NCAlgebra so you might want to try our latest beta version from here, which you can install as a paclet.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.