It is Easter Sunday when I writing this note. In Getting real we met the Cayley transform and its inverse.
Namely we have defined the unitary matrix
(1)
(2)
We will use the matrix in two ways: either for implementing a similarity transformation
resp.
, or for implementing fractional linear transformation of the type
(3)
In each case a factor, such as in front of the matrix is not important. It will cancel out. Two proportional matrices implement the same similarity transformation and the same fractional linear transformation. We have chosen the factor so as to have
unitary of determinant 1, but that fact will play no role. What is important is the internal structure of
In Getting real we have found that the similarity transformation
transforms complex SU(1,1) matrices into real SL(2,R) matrices. Let us now check the fractional linear transformation implemented by For a general case it is convenient to denote the fractional linear transformation (3) as
From Eq. (3) it can be easily verified that, whenever the results are finite, we have
(4)
One could extend the domain and the range of the transformation by replacing the complex plane by its one-point compactification, the Riemann sphere, but we will not need such an extension.
For us the crucial observation is that the transformation maps the unit disk
onto the upper half-plane
that is onto the set of complex numbers with positive imaginary part. To see this let us examine the properties of
defined by
(5)
The imaginary part of is evidently positive when
It becomes zero when
Conversely, the transformation
maps the upper half-plane
in
To see that this is indeed the case let us calculate
(6)
Now therefore if
then
(7)
Moreover, if that is if
is on the real axis, then its image
is on the unit circle.
We finish this Eater post with an ornament. Consider complex numbers of the form
where
are real integers with
and
Apply the Cayley transform to each such number and plot the number
as a point in the disk. Of course we have to restrict the size of
say to
The result is the following Easter Ornament:

Czy ta różnica między między wschodem a zachodem jest ok?
(przy okazji – jak z polskimi znakami? Używać, czy lepiej nie?)
Może gdyby ograniczenia na m,n,p,q były inne to tej różnicy by nie było.
Tak, jest to wynik ograniczenia na iloczyn pq. Bez tego ograniczenia jajo byłoby symetryczne, Wschód wygladałby tak samo jak Zachód.
Załapane. Dzięki.
It becomes zero when
proof needed.
If
with
real, then
is the real part and
the imaginary part of
.
Therefore
Therefore, apart of the point
which is singular for this transformation, we are ok.
Does it answer your question?
Yes. Thanks.