How can I redefine multiplication this way?

Question

I want the product of matrix objects to be evaluated to the matrix object of dot products of the arguments, but only if the matrices commute. Why this does not work?

Unprotect[Times];
Matrix[x_] Matrix[y_] ^= Matrix[x . y] /; x . y == y . x  ;
Protect[Times];

Answer 1

If you want to define "Matrix" as a new Atom, your approach works.

E.g. let us define matrices a0 and a1 that commute:

a0 = RandomReal[{-1, 1}, {2, 2}];
a1 = a0 . a0;

and also a matrix that does not commute:

b = RandomReal[{-1, 1}, {2, 2}];

If we now define "Matrix":

Clear[Matrix]
Unprotect[Times];
Matrix[x_] Matrix[y_] /; x . y == y . x ^= Matrix[x . y];
Protect[Times];

We get for the product of 2 commuting matrices:

Matrix[a0] Matrix[a1]
(* Matrix[{{0.090018, -0.15005}, {-0.276659, 0.0619167}}] *)

and for non-commuting matrices:

Matrix[a0] Matrix[b]
(* Matrix[{{-0.725916, -0.626387}, {-0.553722, -0.871338}}] \
Matrix[{{0.114884, -0.425647}, {-0.784798, 0.0351695}}] *)