Thanks David,

I will write you directly as you suggested, but for the sake of this thread I wanted to thank you for the good advice and try to clarify what I am looking for.

Example

I want to present to reader with a table of the product of the Pauli matrices like

Table[PauliMatrix[i].PauliMatrix[j] // MatrixForm, {i, 3}, {j, 
   3}] // Grid

I would go on to show the results in terms of the Pauli matrices as well, i.e.

Grid[{{s0, I s3, I s2}, {-I s3, s0, I s1}, {-I s2, -I s1, s0}}]

(with s0 being the 2x2 Identity matrix, and si the i'th Pauli matrix).

Clearly, I can do this by hand, but then I might as well work in a typesetting program (like LaTeX). Also as I advance to to work on Dirac's Gamma matrices, (anti-)commute relations, contractions and such, it would be helpful to have Mathematica recognise e.g. that [s1,s2] = 2 I s3 without necessary being aware of it myself.

The "magic" command would be something like ExpressInTermsOf[myExpression,myFavoriteMatrixList], looking in myExpression for possible ways to rewrite it using the elements in myFavoriteMatrixList.

So I am very curious to see what your FactorOut does.

/Mogens

In fact, giving it a little more thought, this is a rather general request. There are numerous situations where you would want to reduce your expressions by substituting parts of them by named (sub-)expressions, or rather: where you would want to use Mathematica to explore the possibility too see if it leads to a simpler result. Using the above example with

a = x+3

rewriting x+4 as a+1 or identify Exp[3] Exp[x] as Exp[a] are just a couple of other situations.

So I search for a way to tell Mathematica to look for certain patterns that I provide in an expression, and show what the expression would look like if the pattern were to be "eliminated". I realise that in some instances this would require a hand-held effort on my side, because the would be ambiguities, but in the example with the Pauli Matrices I think is pretty straight forward.

Hope some of you experienced users can point me in the right direction.

/Mogens

You should understand that evaluation in the Wolfram Language means something like I PauliMatrix[3] will automatically become an explicit matrix {I, 0}, {0, -I}}. There are ways to preclude this using Hold but they are more trouble than they are worth. An alternative route might be to work with your own purely symbolic pauliMatrix (note lower casing of first word, to distinguish from the built-in function). When you want to do symbolic manipulations you might use NonCommutativeMultiply as operator, and only when you want to convert to explicit matrices do you use substitutions such as {NonCommutativeMultiply->Dot,pauliMatrix->PauliMatrix}.

Also for the purpose at hand you might want to consider introducing a commutator, so that the comcommutative products can be "canonicalized" prior to using a substitution such as this: pauliMatrix[1] ** pauliMatrix[2] -> I*pauliMatrix[3]. One way to go about this is shown here.

https://www.researchgate.net/publication/262223580_Symbolic_FAQ

See the section "Some noncommutative algebraic manipulation", in particular the example that defines and uses a function called ncTimes.

PauliMatrix[1].PauliMatrix[2] and I PauliMatrix[3] evaluate to the same matrix.

In[1119]:= {PauliMatrix[1].PauliMatrix[2], I PauliMatrix[3]}

(* Out[1119]= {{{I, 0}, {0, -I}}, {{I, 0}, {0, -I}}} *)

Given that, it is quite unclear what is wanted here.

On a separate note, you are correct that Solve does not work with the Matrices[...] domains (It works fine with explicit matrices but that is not the same thing).