Cayley transform for Easter

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) Its inverse being given by

(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: