Non Commutative Multiply

• To: mathgroup at smc.vnet.net
• Subject: [mg128593] Non Commutative Multiply
• From: Rodrigo AP <digorna at gmail.com>
• Date: Thu, 8 Nov 2012 02:09:06 -0500 (EST)
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• Delivered-to: l-mathgroup@wolfram.com
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Dear group,

I want to multiply two matrices, for example,

A = {{e, f}, {g, h}}
B = {{a, b}, {c, d}}

Using A.B, Mathematica returns

{{a e + b g, a f + b h}, {c e + d g, c f + d h}}

I would like to get, however, the following result:

{{e a + f c, e b + f d}, {g a + h c, g b + h d}}

since, for me, the entries {a,b,c,d,e,f,g,h} are operators, i.e., they
are non-commutative.

I could solve this problem clearing the attribute "Orderless" in the
built-in function Times:

ClearAttributes[Times, Orderless]

I know, however, this can be dangerous. I tried to define a function
Times2[a_,b_]:=Times[a,b]
and then use ClearAttributes[Times2, Orderless] but it doesn't work.

How could I solve this problem?