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MathGroup Archive 2006

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Re: matrix substitution

  • To: mathgroup at smc.vnet.net
  • Subject: [mg67499] Re: matrix substitution
  • From: Roger Bagula <rlbagula at sbcglobal.net>
  • Date: Wed, 28 Jun 2006 03:52:35 -0400 (EDT)
  • References: <e665nv$n43$1@smc.vnet.net> <e7qmvn$6q8$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Here we have a octonion like D4 like matrix group that should be orthogonal!
Strange off diagonal term :
2*I*(x[2] x[5] - x[1] x[6] + x[4] x[7] - x[3] x[8])
I made up the vector matrix and it's conjugate and looked at the product.
It other truly orthogonal matrix groups the off diagonal terms are zero.
This stange term appears as a kind of implicit between the variables
that has to be zero for the matrix group to be othrogonal in an eight space.
A comment in that the SU(3) 6by6 can be treated by this {1,I} same 
substitution method to give
a 16 element group that would appear to be more D6 like than D4 like.
The first thought is that octonions are like SU(3) is pretty obviously 
wrong looking at these matrices.
Mathematica:
ss[n_] := If[n â?¤ 4, v[[n]], w[[n - 4]]]
g = Sum[x[i]*ss[i], {i, 1, 8}]
gs = x[1]*ss[1] +
x[6]*ss[6] - Sum[x[i]*ss[i], {i, 2, 5}] - Sum[x[i]*ss[i], {i, 7, 8}]
FullSimplify[ExpandAll[g.gs]]
MatrixForm[%]
FullSimplify[ExpandAll[g.g]]


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