Cayley transform

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

:

Image:Cayley transform in complex plane.png

On the upper half of the complex plane, the Cayley transform is: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}}Erwin Kreyszig (1983) Advanced Engineering Mathematics, 5th edition, page 611, Wiley {{ISBN|0471862517}}

:$f(z)\; =\; \backslash frac\; \{z\; -\; i\}\{z\; +\; i\}\; .$

Since $\backslash \{\backslash infty,\; 1,\; -1\backslash \}$ is mapped to $\backslash \{1,\; -i,\; i\backslash \}$, and Möbius transformations permute the generalised circles in the complex plane, $f$ maps the real line to the unit circle. Furthermore, since $f$ is a homeomorphism and $i$ is taken to 0 by $f$, the upper half-plane is mapped to the unit disk.

In terms of the 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 reactance half-plane to the Smith chart used for impedance matching of transmission lines.

In the four-dimensional space of quaternions $a+b\backslash vec\{i\}+c\backslash vec\{j\}+d\backslash vec\{k\}$, the versors

:$u(\backslash theta,\; r)\; =\; \backslash cos\; \backslash theta\; +\; r\; \backslash sin\; \backslash theta$ form the unit 3-sphere.

Since quaternions are non-commutative, elements of its projective line have homogeneous coordinates written $U[a,b]$ to indicate that the homogeneous factor multiplies on the left. The quaternion transform is

:

The real and complex homographies described above are instances of the quaternion homography where $\backslash theta$ is zero or $\backslash pi/2$, respectively.

Evidently the transform takes $u\backslash to\; 0\backslash to\; -1$ and takes $-u\; \backslash to\; \backslash infty\; \backslash to\; 1$.

Evaluating this homography at $q=1$ maps the versor $u$ into its axis:

:$f(u,1)\; =(1+u)^\{-1\}(1-u)\; =\; (1+u)^*(1-u)/\; |1+u|^2.$

But $|1+u|^2\; =\; (1+u)(1+u^*)\; =\; 2\; +\; 2\; \backslash cos\; \backslash theta\; ,\backslash quad\; \backslash text\{and\}\backslash quad\; (1+u^*)(1-u)\; =\; -2\; r\; \backslash sin\; \backslash theta\; .$

Thus $f(u,1)\; =\; -r\; \backslash frac\; \{\backslash sin\; \backslash theta\}\{1\; +\; \backslash cos\; \backslash theta\}\; =\; -r\; \backslash tan\; \backslash frac\{\backslash theta\}\{2\}\; .$

In this form the Cayley transform has been described as a rational parametrization of rotation: Let $t=\backslash tan\backslash phi/2$ in the complex number identitySee Tangent half-angle formula

:$e^\{-i\; \backslash varphi\}\; =\; \backslash frac\{1\; -\; ti\}\{1\; +\; ti\}$

where the right hand side is the transform of $ti$ and the left hand side represents the rotation of the plane by negative $\backslash phi$ radians.

Let $u^*\; =\; \backslash cos\; \backslash theta\; -\; r\; \backslash sin\; \backslash theta\; =\; u^\{-1\}\; .$ Since

:

where the equivalence is in the projective linear group over quaternions, the inverse of $f(u,1)$ is

:

Since homographies are bijections, $f^\{-1\}\; (u,1)$ maps the vector quaternions to the 3-sphere of versors. As versors represent rotations in 3-space, the homography $f^\{-1\}$ produces rotations from the ball in $\backslash R^3$.

Among n×n square matrices over the reals, with I the identity matrix, let A be any skew-symmetric matrix (so that A^T = −A).

Then I + A is invertible, and the Cayley transform

:$Q\; =\; (I\; -\; A)(I\; +\; A)^\{-1\}\; \backslash ,\backslash !$

produces an orthogonal matrix, Q (so that Q^TQ = I). The matrix multiplication in the definition of Q above is commutative, so Q can be alternatively defined as $Q\; =\; (I\; +\; A)^\{-1\}(I\; -\; A)$. 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

:$A\; =\; (I\; -\; Q)(I\; +\; Q)^\{-1\}\; \backslash ,\backslash !$

is a skew-symmetric matrix. (See also: 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

:$Q\; =\; \backslash bigl((I\; -\; A)(I\; +\; A)^\{-1\}\backslash bigr)^2$

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

:$Q\; =\; E(I\; -\; A)(I\; +\; A)^\{-1\}$

for some skew-symmetric matrix A and some diagonal matrix E with ±1 as entries.{{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 }}

A slightly different form is also seen,{{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}}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 requiring different mappings in each direction,

:$\backslash begin\{align\}$

Q &= (I - A)^{-1}(I + A), \\[5mu]

A &= (Q - I)(Q + I)^{-1}.

\end{align}

The mappings may also be written with the order of the factors reversed;{{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, §7.2Howard Eves (1966) Elementary Matrix Theory, § 5.4A Cayley’s Construction of Real Orthogonal Matrices, pages 365–7, Allyn & Bacon however, A always commutes with (μI ± A)⁻¹, so the reordering does not affect the definition.

In the 2×2 case, we have

:$$

\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} .

The 180° rotation matrix, −I, is excluded, though it is the limit as tan ^θ⁄₂ goes to infinity.

In the 3×3 case, we have

:$$

\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} ,

where K = w² + x² + y² + z², and where w = 1. This we recognize as the rotation matrix corresponding to quaternion

:$w\; +\; \backslash mathbf\{i\}\; x\; +\; \backslash mathbf\{j\}\; y\; +\; \backslash mathbf\{k\}\; z\; \backslash ,\backslash !$

(by a formula Cayley had published the year before), except scaled so that w = 1 instead of the usual scaling so that w² + x² + y² + z² = 1. Thus vector (x,y,z) is the unit axis of rotation scaled by tan ^θ⁄₂. Again excluded are 180° rotations, which in this case are all Q which are symmetric (so that Q^T = Q).

One can extend the mapping to complex matrices by substituting "unitary" for "orthogonal" and "skew-Hermitian" for "skew-symmetric", the difference being that the transpose (·^T) is replaced by the conjugate transpose (·^H). 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 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,

:$$

\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} .

Note that A is skew-symmetric (respectively, skew-Hermitian) if and only if Q is orthogonal (respectively, unitary) with no eigenvalue −1.

An infinite-dimensional version of an inner product space is a Hilbert space, and one can no longer speak of matrices. However, matrices are merely representations of linear operators, 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}}

:$\backslash begin\{align\}$

U &{}= (A - \mathbf{i}I) (A + \mathbf{i}I)^{-1} \\

A &{}= \mathbf{i}(I + U) (I - U)^{-1}

\end{align}

Here the domain of U, dom U, is (A+iI) dom A. See self-adjoint operator for further details.

• Bilinear transform
• Extensions of symmetric operators

{{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

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| volume=32

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| 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. 332–336) in {{Citation

| last=Cayley

| first=Arthur

| author-link=Arthur Cayley

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| isbn=

| 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
• {{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}}

Category:Conformal mappings

Category:Transforms

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