From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Subject: Re: What is a Cayley-Mobius transform?
Date: 29 Jan 2000 09:04:38 GMT
Newsgroups: sci.math
Summary: [missing]

In article <388EFA52.5DB4DA33@psy.uva.nl>,
Raoul Grasman  <grasman@psy.uva.nl> wrote:
>I was reading this paper on (the history of) spectral estimation, in
>which the relation is explained between the spectral representation
>theorem of Von Neumann of Hermitian operators defined on a Hilbert space
>and the generalized harmonic analysis of Wiener. This relation is
>established by the so-called Cayley-Moebius transform, which said to be
>one-to-one mapping Hermitian operators onto unitary operators. Can
>anyone explain to me how this Cayley-Moebius transform is defined, what
>it means/implies and where I can find some readable (say undergraduate
>level) text on it?

I have to say I hadn't heard this combined terminology before but a scan of 
MathSciNet suggests this is supposed to be what I know as the Mobius
transform. Here's a reformatted version of the review    95i:65109 

   Garratt, T. J.; Moore, G.(4-LNDIC); Spence, A.(4-BATH)
   A generalised Cayley transform for the numerical detection of Hopf
   bifurcations in large systems. 
   Contributions in numerical mathematics, 177--195,
   World Sci. Ser. Appl. Anal., 2, World Sci. Publishing, River Edge, NJ, 1993.

   The authors deal with the stability analysis of steady states,
   especially with the detection of Hopf bifurcation points, in
   high-dimensional systems of parameter-dependent nonlinear differential
   equations: dx/dt + f(x,\lambda)=0,  f:R^{n+1} -> R^n,  x\in R^n, 
   \lambda\in R. The computation of the whole spectrum of  A := f_x (which is
   the standard tool to determine stability) is too expensive for large  n .
   Therefore it is proposed to compute iteratively some of the
   eigenvalues with smallest real part (which determine stability).
   Appropriate methods, like subspace iteration or Arnoldi's method,
   converge to extremal parts of the spectrum with respect to the
   modulus. To use them to approximate the eigenvalues with smallest real
   part, they have to be performed on a transform  C(A)  of  A  which
   maps the wanted eigenvalues of  A  bijectively to those with largest
   real parts of  C(A).  The authors choose the so-called
   generalized Cayley transform:  C(A;\alpha_1,\alpha_2) :=
           ( A - \alpha_1 I )^{-1} ( A - \alpha_2 I ),   
   \alpha_1 < \alpha_2,  \alpha_1  not an eigenvalue of  A.  This is
   nothing other than the well-known Mobius transform for matrices.
   Obviously the eigenvalues of   A  are transformed by the
   corresponding Mobius transformation. Using the mapping properties of
   the Mobius transformation, one can manage to separate the wanted
   eigenvalues of the transform  C(A)  with respect to the modulus.
   Therefore, iterative methods can be used successfully.

   The authors describe extensively the mapping properties of the Mobius
   transformation and the corresponding relation to the convergence
   behavior of the iterative solvers. Furthermore, numerical details and
   difficulties of the method in the continuation and several strategies
   are discussed. The results are summarized in algorithmic form.
   Finally, test results using a model of a tubular reactor are appended.

                         Reviewed by Qinghua Zheng

   (c) Copyright American Mathematical Society 2000