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Commuting Matrices

• To: mathgroup at smc.vnet.net
• Subject: [mg24541] Commuting Matrices
• From: Vishnu Jejjala <jejjala at ux8.cso.uiuc.edu>
• Date: Mon, 24 Jul 2000 03:04:20 -0400 (EDT)
• Organization: University of Illinois at Urbana-Champaign
• Sender: owner-wri-mathgroup at wolfram.com

I have a collection of matrices which satisfy a matrix algebra.
Call three of the matrices Phi1, Phi2, and Phi3.  I want to take
expressions of the form (Phi1.Phi2.Phi3)^n and `normal order' them
using the commutation relations of the algebra so that the terms
in the expansion are of the form Phi1^a.Phi2^b.Phi3^c.  Is there
a slick way to do this?

My brute force method of using replaces isn't working.  One of
the problems I run into is that when I obtain an expression like
Phi3.(k Phi1.Phi3).Phi2.Phi3, I can't seem to teach Mathematica
to realize that the k is a scalar and factors through and that it
should apply the commutation rules to k Phi3.Phi1.Phi3.Phi2.Phi3.

Perhaps someone can offer suggestions.  Thanks.

--Vishnu