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From: JEMebius on 29 Jul 2010 07:40
carlos(a)colorado.edu wrote:
> This hopefully makes my previous post more explicit.
> Consider the 4 x 4 matrix written in Mathematica notation,
>
> T= {{1+c2,s2,s2,1-c2},
> {-s2,1+c2,-1+c2,s2},
> {-s2,-1+c2,1+c2,s2},
> {1-c2,-s2,-s2,1+c2}}/2;
>
> in which c2=Cos[2*phi], s2=Sin[2*phi] and phi is an arbitrary angle.
> The 4 eigenvalues lie on the unit circle. Since the transpose is not
> equal
> to the inverse except for specific angles, T is not orthogonal
> (it can be mapped, however, to an orthogonal matrix by a diagonal
> similarity transformation independent of phi).
>
> I was going to call T a "unimodular" matrix but found the name
> taken by graph theorists. What would be an appropriate name,
> "S-orthogonal", "Schur-Cohn" ... ?
>

I decided to check matrix T by means of a Maple run. Changed the notation somewhat:
"c2" to "c"; "s2" to "s"; in my Maple source I have finally

A :=
array ([[1+c, s, s, 1-c], [-s, 1+c, -1+c, s], [-s, -1+c, 1+c, s], [1-c, -s, -s, 1+c]]);

instead of the original matrix T.

Bottom line: matrix T is a 4D rotation matrix for all values of angle phi,
so somewhere in your calculations something must have gone wrong.

Please find Maple source, Maple-6 run, Maple-9.5 run and screenshots at
http://www.xs4all.nl/~jemebius/Math.htm ,
reachable from my home page at http://www.xs4all.nl/~jemebius/ .

All the best and happy studies: Johan E. Mebius

From: carlos on 29 Jul 2010 14:37
On Jul 29, 5:40 am, JEMebius <jemeb...(a)xs4all.nl> wrote:
> car...(a)colorado.edu wrote:
> > This hopefully makes my previous post more explicit.
> > Consider the 4 x 4 matrix written in Mathematica notation,
>
> >    T= {{1+c2,s2,s2,1-c2},
> >         {-s2,1+c2,-1+c2,s2},
> >         {-s2,-1+c2,1+c2,s2},
> >         {1-c2,-s2,-s2,1+c2}}/2;
>
> > in which c2=Cos[2*phi], s2=Sin[2*phi] and phi is an arbitrary angle..
> > The 4 eigenvalues lie on the unit circle.  Since the transpose is not
> > equal
> > to the inverse except for specific angles, T is not orthogonal
> > (it can be mapped, however, to an orthogonal matrix by a diagonal
> > similarity transformation independent of phi).
>
> > I was going to call T a "unimodular" matrix but found the name
> > taken by graph theorists.  What would be an appropriate name,
> > "S-orthogonal", "Schur-Cohn" ... ?
>
> I decided to check matrix T by means of a Maple run. Changed the notation somewhat:
> "c2" to "c"; "s2" to "s"; in my Maple source I have finally
>
> A :=
> array ([[1+c, s, s, 1-c], [-s, 1+c, -1+c, s], [-s, -1+c, 1+c, s], [1-c, -s, -s, 1+c]]);
>
> instead of the original matrix T.
>
> Bottom line: matrix T is a 4D rotation matrix for all values of angle phi,
> so somewhere in your calculations something must have gone wrong.
>
> Please find Maple source, Maple-6 run, Maple-9.5 run and screenshots athttp://www.xs4all.nl/~jemebius/Math.htm,
> reachable from my home page athttp://www.xs4all.nl/~jemebius/.
>
> All the best and happy studies: Johan E. Mebius

Yes, it is the rotation matrix for a 4-pseudovector that comes
up in the search for invariants of high order tensor
derivatives. To further clarify:

dT=T^(-1)-T'={{0,-s2,-s2,0},{s2,0,0,-s2},{s2,0,0,-s2},{0,s2,s2,0}}

so inverse does NOT equal transpose unless phi=n*Pi, n=0,1,..
However, T.T'=I since dT is antisymmetric.

Here is a 8-dimensional version found by Mathematica:

T={{3+4*c2+c4,2*s2+s4,1-c4,
2*s2-s4,2*s2+s4,1-c4,2*s2-s4,
3-4*c2+c4},{-3*(2*s2+s4),1+4*c2+3*c4,2*s2+3*s4,
3-3*c4,3*(-1+c4),-2*s2+3*s4,-1+4*c2-3*c4,
6*s2-3*s4},{3-3*c4,-2*s2-3*s4,1+4*c2+3*c4,3*(2*s2+s4),
6*s2-3*s4,1-4*c2+3*c4,-2*s2+3*s4,3-3*c4},{-2*s2+s4,
1-c4,-2*s2-s4,3+4*c2+c4,-3+4*c2-c4,2*s2-s4,-1+c4,
2*s2+s4},{-2*s2-s4,-1+c4,-2*s2+s4,-3+4*c2-c4,
3+4*c2+c4,2*s2+s4,1-c4,2*s2-s4},{3-3*c4,2*s2-3*s4,
1-4*c2+3*c4,-6*s2+3*s4,-3*(2*s2+s4),1+4*c2+3*c4,
2*s2+3*s4,3-3*c4},{-6*s2+3*s4,-1+4*c2-3*c4,2*s2-3*s4,
3*(-1+c4),3-3*c4,-2*s2-3*s4,1+4*c2+3*c4,
3*(2*s2+s4)},{3-4*c2+c4,-2*s2+s4,1-c4,-2*s2-s4,
-2*s2+s4,1-c4,-2*s2-s4,3+4*c2+c4}}/8;

with c2=Cos[2*phi],s2=Sin[2*phi],c4=Cos[4*phi],s4=Sin[4*phi].
Process can be continued to any number of dimensions. Once T
is found, invariants can be obtained by solving a homogeneous form
of the discrete-time Sylvester equation. No problems with the
method, just of nomenclature anticipating reviewers' comments.



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