Problems with definition of a function with matrices

Question

I want to define the product between two matrices. If I don't define it as function everything is fine (lines at the end of the image), but when I try to define it as a function it gets strange things. What is going on? And how do I solve it? The first one is

Okubo[x03_,y03_] = \[Omega]*(x03.y03) - \[Omega]*\[Omega] (y03.x03) - (\[Omega] - \[Omega]*\[Omega])/3 Tr[x03.y03] IdentityMatrix [3];

Okubo[e, e] // MatrixForm

while the second one (working properly) is

\[Omega]*(e.e) - \[Omega]*\[Omega] (e.e) - (\[Omega] - \[Omega]*\[Omega])/3 Tr[e.e] IdentityMatrix [3] 

The lement "e" is the the one on which I'm testing the thing and is

e = ( { {2, 0, 0},{0, -1, 0}, {0, 0, -1}} );

Answer 1

Use :=

Clear["Global`*"]
e = ({{2, 0, 0}, {0, -1, 0}, {0, 0, -1}});
Okubo[x03_,y03_] := ω*(x03 . y03) - ω*ω (y03.x03) - (ω - ω*ω)/3 
     Tr[x03.y03] IdentityMatrix[3];

Okubo[e, e] // MatrixForm

(ω*(e . e) - ω*ω (e.e) - (ω - ω*ω)/3 Tr[e . e] IdentityMatrix[3]) // MatrixForm

The reason is, when you used immediate assignment, then Mathematica evaluated the body of the function immediately. This resulted in

Okubo[x03_,y03_] = ω*(x03 . y03) - ω*ω (y03 . x03) - (ω - ω*ω)/3 
    Tr[x03 . y03] IdentityMatrix[3]

Which is not what you want. Because it did not know that x03 and y03 were matrices. But with delayed assignments, it did the right . operation since the arguments are now known to be matrices.

It is always best to define functions using delayed assignment unless there is specific reason not to.