| Try viewing your matrix, A, as representing a linear transformation on a
two dimensional vector space over the reals.
For A**n to equal I, an eigenvalue of A must be the n'th root of unity.
Representing A as an ABCD matrix,
| a b |
A = | |
| c d |
The constraints on its elements are:
2 2 2 2
4*cos(2*pi/n) = a + 2*a*d + d = (a + d)
2 2 2
4*sin(2*pi/n) = 4*(a*d-b*c) - (a + d) = 4*Det(A) - Tr(A)
1 = (a*d - b*c)
These are clearly satisfied in the examples given. Since the determinant
of A is the volume measure of the transformation it must equal 1.
- Jim
|
| Here's another way of looking at this problem.
Consider the group of Moebius (bilinear) transformations on the
complex plane:
az + b
w(z) = --------
cz + d
We can identify composition of these transformations with multiplication
of the matrices:
| a b |
A = | |
| c d |
If Det(A) vanishes, then w(z) reduces to a constant, and is thus non
invertable, as A would be.
Now the problem is to look for periodic sequences of bilinear transformations,
and we can generalize A to be complex.
I believe that its known this is only true for n = 1, 2, 3, 4, and 6
when the elements of A are rational. This is related to the fact that
the characteristic equation of A is a quadratic, and only the numbers
n = 2, 3, 4, and 6 have 2 or fewer relatively prime integers less than n.
(Euler's function, phi(n) .le. 2). [In my prior note, I should have
referred to 'primitive root of unity of order n' for the eigenvalue of A,
but the idea was right.]
If you stereographically project the complex plane onto the Riemann sphere,
perform some rigid motion of the sphere, and project back I think this is
equivalent to a bilinear transformation; it might be possible to identify
symmetry groups of convex polyhedra on the sphere with periodic sequences
of these transformations but that's just a guess.
Bilinear transformations are familiar to me from microwave and RF filter
design. The impedance or admittance of a network is related to its
scattering parameters via a bilinear transform. If the network is
lossless, the S matrix is unitary; this is one key in the synthesis
of a network transfer function on an insertion loss basis. You can
map periodic structures (transmission lines, sampled digital filters) onto
continuous filters via a bilinear transform (the tan(theta/2) substitution)
and adapt classical filter designs to these areas. This is where the
Smith chart comes from. Since circles map to circles its a powerful
design aid since you can plot noise circles and stability circles on
the Smith chart and visualize the boundries of behaviour of an amplifier.
There are network analyzers that directly plot S parameters for you.
Well, so much for my so called 'knowledge' of mathematics...
Check out Schwerdtfeger's book "The geometry of complex numbers" for
more goodies related to this area (one of those inexpensive Dover reprints).
- Jim
|