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From: JEMebius on 31 Jul 2010 18:44
carlos(a)colorado.edu wrote:
> 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 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
>
> 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.
>
>
>
I would much like to know how these matrices arise in connexion with tensor derivatives.
Please could you give some literature? - thanks in advance!

I was wrong with my statement that "eigenvalues all on the unit circle" implies that the
matrix is orthogonal. I just thought of orthogonal similarity transformations only, not of
general similarity transformations.

Believe it or not: up to yesterday I never did the pencil-and-paper work in the complex
eigenvalues and eigenvectors of 2D rotation matrices; in algebra applied to geometry I am
rather real-inclined than complex-inclined. So after some fifty-odd years I finally
plugged a hole in my math education.

Now returning to your original matrix T. I did a second Maple job, this time with the
original matrix T copied straightaway from your original post and pasted into an empty
Maple worksheet.
See the newly added screenshots on http://www.xs4all.nl/~jemebius/Math.htm .
Again one sees that T is orthogonal for all values of Phi.

Matrix T as a function of Phi turns out to be an interesting 4D rotation. With the help of
a home-made 4D matrix calculator program this rotation was decomposed into its left- and
right-isoclinic factors, after which I enjoyed much what went on in my 4D graphics
animation program.


To be specific, T is the product of the left-isoclinic rotation

TL = array([[c, s, 0, 0], [-s, c, 0, 0], [0, 0, c, s], [0, 0, -s, c]])

and the right-isoclinic rotation

TR = array([[c, 0, s, 0], [0, c, 0, s], [-s, 0, c, 0], [0, -s, 0, c]]),

where c = cos(Phi) and S = sin(Phi).


The matrix of the six angular velocities is (in Maple notation, i.e. row-by-row)

W = array([[0, 1, 1, 0], [-1, 0, 0, 1], [-1, 0, 0, 1], [0, -1, -1, 0]]),

which means that with a Cartesian coordinate system OUXYZ we have rotations from OX to OU,
from OY to OU, from OZ to OX and from OZ to OY with equal angular velocities.


In general: W = array([[0, Wxu, Wyu, Wzu], [Wux, 0, Wyx, Wzx], [Wuy, Wxy, 0, Wzy], [Wuz,
Wxz, Wyz, 0]]),
where Wij (i, j = u, x, y, z) means the angular velocity in the coordinate plane Oij,
reckoned positive if rotating from half-axis +i to half-axis +j, and negative if rotating
from half-axis +j to half-axis +i.
W is antisymmetric, so one need specify six values only, for instance Wux, Wuy, Wuz, Wyz,
Wzx, Wxy. This is the convention in my 4D geometry software.


Ciao: Johan E. Mebius
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