Editing Cayley transform

{{Short description|Mathematical operation}}
In [[mathematics]], the '''Cayley transform''', named after [[Arthur Cayley]], is any of a cluster of related things. As originally described by {{Harvtxt|Cayley|1846}}, the Cayley transform is a mapping between [[skew-symmetric matrix|skew-symmetric matrices]] and [[special orthogonal matrix|special orthogonal matrices]]. The transform is a [[homography]] used in [[real analysis]], [[complex analysis]], and [[quaternionic analysis]]. In the theory of [[Hilbert space]]s, the Cayley transform is a mapping between [[linear operator]]s {{Harv|Nikolski|1988}}.

==Real homography==
A simple example of a Cayley transform can be done on the [[real projective line]]. The Cayley transform here will permute the elements of {1, 0, −1, ∞} in sequence. For example, it maps the [[positive real numbers]] to the interval [−1, 1]. Thus the Cayley transform is used to adapt [[Legendre polynomials]] for use with functions on the positive real numbers with [[Legendre rational functions]].

As a real [[homography]], points are described with [[projective coordinates]], and the mapping is
:<math>[y,\ 1] = \left[\frac {x - 1}{x +1},\ 1\right] \thicksim [x - 1, \ x + 1] = [x,\ 1]\begin{pmatrix}1 & 1 \\ -1 & 1 \end{pmatrix} .</math>

==Complex homography==
[[Image:Cayley transform in complex plane.png|thumb|right| 300px|Cayley transform of upper complex half-plane to unit disk]]
On the [[Upper half-plane|upper half]] of the [[complex plane]], the Cayley transform is:<ref>Robert Everist Green & [[Steven G. Krantz]] (2006) ''Function Theory of One Complex Variable'', page 189, [[Graduate Studies in Mathematics]] #40, [[American Mathematical Society]] {{ISBN|9780821839621}}</ref><ref>[[Erwin Kreyszig]] (1983) ''Advanced Engineering Mathematics'', 5th edition, page 611, Wiley {{ISBN|0471862517}}</ref>
:<math>f(z) = \frac {z - i}{z + i} .</math>
Since <math>\{\infty, 1, -1\}</math> is mapped to <math>\{1, -i, i\}</math>, and [[Möbius transformation]]s permute the [[generalised circle]]s in the [[complex plane]], <math>f</math> maps the real line to the [[unit circle]]. Furthermore, since <math>f</math> is a [[homeomorphism]] and <math>i</math> is taken to 0 by <math>f</math>, the upper half-plane is mapped to the [[unit disk]].

In terms of the [[mathematical model|models]] of [[hyperbolic geometry]], this Cayley transform relates the [[Poincaré half-plane model]] to the [[Poincaré disk model]].

In electrical engineering the Cayley transform has been used to map a [[electrical reactance|reactance]] half-plane to the [[Smith chart]] used for [[impedance matching]] of transmission lines.

==Quaternion homography==
In the [[four-dimensional space]] of [[quaternion]]s <math>a+b\vec{i}+c\vec{j}+d\vec{k}</math>, the [[versor]]s
:<math>u(\theta, r) = \cos \theta + r \sin \theta </math> form the unit [[3-sphere]].

Since quaternions are non-commutative, elements of its [[projective line over a ring|projective line]] have homogeneous coordinates written <math>U[a,b]</math> to indicate that the homogeneous factor multiplies on the left. The quaternion transform is
:<math>f(u,q) = U[q,1]\begin{pmatrix}1 & 1 \\ -u & u \end{pmatrix} = U[q - u,\ q + u] \sim U[(q + u)^{-1}(q - u),\ 1].</math>

The real and complex homographies described above are instances of the quaternion homography where <math>\theta</math> is zero or <math>\pi/2</math>, respectively.
Evidently the transform takes <math>u\to 0\to -1</math> and takes <math>-u \to \infty \to 1</math>.

Evaluating this homography at <math>q=1</math> maps the versor <math>u</math> into its axis:
:<math>f(u,1) =(1+u)^{-1}(1-u) = (1+u)^*(1-u)/ |1+u|^2.</math>
But <math>|1+u|^2 = (1+u)(1+u^*) = 2 + 2 \cos \theta ,\quad \text{and}\quad (1+u^*)(1-u) = -2 r \sin \theta .</math>

Thus <math>f(u,1) = -r \frac {\sin \theta}{1 + \cos \theta} = -r \tan \frac{\theta}{2} .</math>

In this form the Cayley transform has been described as a rational parametrization of rotation: Let <math>t=\tan\phi/2</math> in the complex number identity<ref>See [[Tangent half-angle formula]]</ref>
:<math>e^{-i \varphi} = \frac{1 - ti}{1 + ti} </math>
where the right hand side is the transform of <math>ti</math> and the left hand side represents the rotation of the plane by negative <math>\phi</math> radians.

===Inverse===
Let <math>u^* = \cos \theta - r \sin \theta = u^{-1} .</math> Since
:<math>\begin{pmatrix} 1 & 1 \\ -u & u \end{pmatrix}\ \begin{pmatrix} 1 & -u^*  \\ 1 & u^* \end{pmatrix} \ = \ \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \ \sim \  \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \ ,</math>
where the equivalence is in the [[projective linear group]] over quaternions, the [[inverse function|inverse]] of <math>f(u,1)</math> is
:<math>U[p,1] \begin{pmatrix} 1 & -u^* \\ 1 & u^* \end{pmatrix} \ = \ U[p+1,\ (1-p)u^*] \sim U[u(1-p)^{-1} (p+1), \ 1] .</math>
Since homographies are [[bijection]]s, <math>f^{-1} (u,1)</math> maps the vector quaternions to the 3-sphere of versors. As versors represent rotations in 3-space, the homography <math>f^{-1}</math> produces rotations from the ball in <math>\R^3</math>.

== Matrix map ==
Among ''n''×''n'' [[square matrix|square matrices]] over the [[real number|reals]], with ''I'' the [[identity matrix]], let ''A'' be any [[skew-symmetric matrix]] (so that ''A''<sup>T</sup>&nbsp;= −''A''). 

Then ''I''&nbsp;+&nbsp;''A'' is [[invertible matrix|invertible]], and the Cayley transform
:<math> Q = (I - A)(I + A)^{-1} \,\!</math>
produces an [[orthogonal matrix]], ''Q'' (so that ''Q''<sup>T</sup>''Q''&nbsp;= ''I''). The matrix multiplication in the definition of ''Q'' above is commutative, so ''Q'' can be alternatively defined as <math> Q = (I + A)^{-1}(I - A)</math>. In fact, ''Q'' must have determinant +1, so is special orthogonal. 

Conversely, let ''Q'' be any orthogonal matrix which does not have −1 as an [[eigenvalue]]; then
:<math> A = (I - Q)(I + Q)^{-1} \,\!</math>
is a skew-symmetric matrix.  (See also: [[Involution (mathematics)|Involution]].) The condition on ''Q'' automatically excludes matrices with determinant −1, but also excludes certain special orthogonal matrices.

However, any rotation (special orthogonal) matrix ''Q'' can be written as

:<math>Q = \bigl((I - A)(I + A)^{-1}\bigr)^2</math>

for some skew-symmetric matrix ''A''; more generally any orthogonal matrix ''Q'' can be written as

:<math>Q = E(I - A)(I + A)^{-1}</math>

for some skew-symmetric matrix ''A'' and some diagonal matrix ''E'' with ±1 as entries.<ref>{{cite arXiv |last=Gallier |first=Jean |author-link=Jean Gallier |title=Remarks on the Cayley Representation of Orthogonal Matrices and on Perturbing the Diagonal of a Matrix to Make it Invertible | eprint=math/0606320 |year=2006}}{{pb}}
As described by Gallier, the first of these results is a sharpened variant of {{cite book |last=Weyl |first=Hermann |author-link=Hermann Weyl |year=1946 |title=The Classical Groups |edition=2nd |publisher=Princeton University Press |at=Lemma 2.10.D, p. 60 }}{{pb}}
The second appeared as an exercise in {{cite book |last=Bellman |first=Richard |title=Introduction to Matrix Analysis |publisher=SIAM Publications |year=1960 |at=§6.4 exercise 11, p. 91–92 }}
</ref>

A slightly different form is also seen,<ref>{{Citation | last1=Golub | first1=Gene H. | author1-link=Gene H. Golub | last2=Van Loan | first2=Charles F. | author2-link=Charles F. Van Loan | title=Matrix Computations  | edition=3rd  | publisher=[[Johns Hopkins University Press]] | year=1996 | isbn=978-0-8018-5414-9}}</ref><ref>F. Chong (1971) "A Geometric Note on the Cayley Transform", pages 84,5 in ''A Spectrum of Mathematics: Essays Presented to H. G. Forder'', [[John C. Butcher]] editor, [[Auckland University Press]]</ref> requiring different mappings in each direction,
:<math>\begin{align}
 Q &= (I - A)^{-1}(I + A), \\[5mu]
 A &= (Q - I)(Q + I)^{-1}.
\end{align}</math>

The mappings may also be written with the order of the factors reversed;<ref>{{Citation| last1=Courant| first1=Richard| author1-link=Richard Courant| last2=Hilbert| first2=David| author2-link=David Hilbert| title=Methods of Mathematical Physics| volume=1| edition=1st English| publisher=Wiley-Interscience  | year=1989  | pages=536, 7  | place=New York  | isbn=978-0-471-50447-4}} Ch.VII,&nbsp;§7.2</ref><ref>[[Howard Eves]] (1966) ''Elementary Matrix Theory'', § 5.4A Cayley’s Construction of Real Orthogonal Matrices, pages 365–7, [[Allyn & Bacon]]</ref> however, ''A'' always commutes with (μ''I''&nbsp;±&nbsp;''A'')<sup>−1</sup>, so the reordering does not affect the definition.

=== Examples ===
In the 2×2 case, we have
:<math>
\begin{bmatrix} 0 & \tan \frac{\theta}{2} \\ -\tan \frac{\theta}{2} & 0 \end{bmatrix}
\leftrightarrow
\begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix} .
</math>
The 180° [[rotation matrix]], −''I'', is excluded, though it is the limit as tan&nbsp;<sup>θ</sup>⁄<sub>2</sub> goes to infinity.

In the 3×3 case, we have
:<math>
\begin{bmatrix} 0 & z & -y \\ -z & 0 & x \\ y & -x & 0 \end{bmatrix}
\leftrightarrow
\frac{1}{K}
\begin{bmatrix}
  w^2+x^2-y^2-z^2 & 2 (x y-w z) & 2 (w y+x z) \\
  2 (x y+w z) & w^2-x^2+y^2-z^2 & 2 (y z-w x) \\
  2 (x z-w y) & 2 (w x+y z) & w^2-x^2-y^2+z^2
\end{bmatrix} ,
</math>

where ''K''&nbsp;=&nbsp;''w''<sup>2</sup>&nbsp;+&nbsp;''x''<sup>2</sup>&nbsp;+&nbsp;''y''<sup>2</sup>&nbsp;+&nbsp;''z''<sup>2</sup>, and where ''w''&nbsp;=&nbsp;1. This we recognize as the rotation matrix corresponding to [[quaternion]]

:<math> w + \mathbf{i} x + \mathbf{j} y + \mathbf{k} z \,\!</math>

(by a formula Cayley had published the year before), except scaled so that ''w''&nbsp;= 1 instead of the usual scaling so that ''w''<sup>2</sup>&nbsp;+&nbsp;''x''<sup>2</sup>&nbsp;+&nbsp;''y''<sup>2</sup>&nbsp;+&nbsp;''z''<sup>2</sup>&nbsp;=&nbsp;1. Thus vector (''x'',''y'',''z'') is the unit axis of rotation scaled by tan&nbsp;<sup>θ</sup>⁄<sub>2</sub>. Again excluded are 180° rotations, which in this case are all ''Q'' which are [[symmetric matrix|symmetric]] (so that ''Q''<sup>T</sup>&nbsp;= ''Q'').

=== Other matrices ===
One can extend the mapping to [[complex number|complex]] matrices by substituting "[[unitary matrix|unitary]]" for "orthogonal" and "[[skew-Hermitian matrix|skew-Hermitian]]" for "skew-symmetric", the difference being that the transpose (·<sup>T</sup>) is replaced by the [[conjugate transpose]] (·<sup>H</sup>). This is consistent with replacing the standard real [[inner product]] with the standard complex inner product. In fact, one may extend the definition further with choices of [[Hermitian adjoint|adjoint]] other than transpose or conjugate transpose.

Formally, the definition only requires some invertibility, so one can substitute for ''Q'' any matrix ''M'' whose eigenvalues do not include −1. For example,
:<math>
\begin{bmatrix} 0 & -a & ab - c \\ 0 & 0 & -b \\ 0 & 0 & 0 \end{bmatrix}
\leftrightarrow
\begin{bmatrix} 1 & 2a & 2c \\ 0 & 1 & 2b \\ 0 & 0 & 1 \end{bmatrix} .
</math>
Note that ''A'' is skew-symmetric (respectively, skew-Hermitian) if and only if ''Q'' is orthogonal (respectively, unitary) with no eigenvalue −1.

== Operator map ==
An infinite-dimensional version of an [[inner product space]] is a [[Hilbert space]], and one can no longer speak of [[matrix (mathematics)|matrices]]. However, matrices are merely representations of [[linear operator]]s, and these can be used. So, generalizing both the matrix mapping and the complex plane mapping, one may define a Cayley transform of operators.{{sfn | Rudin | 1991 | p=356-357 §13.17}}
:<math>\begin{align}
 U &{}= (A - \mathbf{i}I) (A + \mathbf{i}I)^{-1} \\
 A &{}= \mathbf{i}(I + U) (I - U)^{-1}
\end{align}</math>
Here the domain of ''U'', dom&nbsp;''U'', is (''A''+'''i'''''I'')&nbsp;dom&nbsp;''A''. See [[self-adjoint operator#Extensions of symmetric operators|self-adjoint operator]] for further details.

== See also ==
* [[Bilinear transform]]
* [[Extensions of symmetric operators]]

== References ==
{{Reflist}}

* Sterling K. Berberian (1974) ''Lectures in Functional Analysis and Operator Theory'', [[Graduate Texts in Mathematics]] #15, pages 278, 281, Springer-Verlag {{ISBN|978-0-387-90081-0}}
* {{Citation
 | last=Cayley
 | first=Arthur
 | author-link=Arthur Cayley
 | year=1846
 | title=Sur quelques propriétés des déterminants gauches
 | journal=[[Journal für die reine und angewandte Mathematik]]
 | volume=32
 | pages=119–123
 | issue=2
 | url=https://www.digizeitschriften.de/id/243919689_0032%7Clog19?tify=%7B%22pages%22%3A%5B129%5D%2C%7D
 | issn=0075-4102
 | doi=10.1515/crll.1846.32.119
}}; reprinted as article 52 (pp.&nbsp;332–336) in {{Citation
 | last=Cayley
 | first=Arthur
 | author-link=Arthur Cayley
 | year=1889
 | title=The collected mathematical papers of Arthur Cayley
 | publisher=[[Cambridge University Press]]
 | volume=I (1841–1853)
 | pages=332–336
 | isbn=<!-- none given -->
 | url=http://www.hti.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath;cc=umhistmath;rgn=full%20text;idno=ABS3153.0001.001;didno=ABS3153.0001.001;view=image;seq=00000349
}}
* Lokenath Debnath & Piotr Mikusiński (1990) ''Introduction to Hilbert Spaces with Applications'', page 213, [[Academic Press]] {{ISBN|0-12-208435-7}}
* Gilbert Helmberg (1969) ''Introduction to Spectral Theory in Hilbert Space'', page 288, § 38: The Cayley Transform, Applied Mathematics and Mechanics #6, [[North Holland (publisher)|North Holland]]
* {{Citation |last=Nikolski |first=Nikolai Kapitonovich |author-link=Nikolai Kapitonovich Nikolski |contribution=Cayley transform |contribution-url=http://www.encyclopediaofmath.org/index.php?title=Cayley_transform |editor-last=Hazewinkel |editor-first= Michiel |editor-link=Michiel Hazewinkel |title=[[Encyclopaedia of Mathematics]] |volume=2 |year=1988 |page=80 |publisher=Kluwer |doi=10.1007/978-94-009-6000-8 |isbn=978-94-009-6002-2 }}; translated from the Russian {{Citation |editor-last=Vinogradov |editor-first=Ivan Matveevich |editor-link=Ivan Vinogradov |title=Matematicheskaya Entsiklopediya |place=Moscow |publisher=Sovetskaya Entsiklopediya |year=1977}}
* Henry Ricardo (2010) ''A Modern Introduction to Linear Algebra'', page 504, [[CRC Press]] {{ISBN|978-1-4398-0040-9}} .
* {{Rudin Walter Functional Analysis|edition=2}} <!-- {{sfn | Rudin | 1991 | p=356-357 §13.17}} -->


[[Category:Conformal mappings]]
[[Category:Transforms]]

[[ru:Преобразование Мёбиуса#Примеры]]