Cayley transform

In mathematics, the Cayley transform, named after Arthur Cayley, is any of a cluster of related things. As originally described by Cayley (1846), the Cayley transform is a mapping between skew-symmetric matrices and special orthogonal matrices. In complex analysis, the Cayley transform is a conformal mapping (Rudin 1987) in which the image of the upper complex half-plane is the unit disk (Remmert 1991, pp. 82ff, 275). And in the theory of Hilbert spaces, the Cayley transform is a mapping between linear operators (Nikol’skii 2001).

Matrix map

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} \,\!

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} \,\!

is a skew-symmetric matrix. The condition on Q automatically excludes matrices with determinant −1, but also excludes certain special orthogonal matrices. Some authors use a superscript "c" to denote this transform, writing Q = A^c and A = Q^c.

This version of the Cayley transform is its own functional inverse, so that A = (A^c)^c and Q = (Q^c)^c. A slightly different form is also seen (Golub & Van Loan 1996), requiring different mappings in each direction (and dropping the superscript notation):

\begin{align} Q &{}= (I - A)^{-1}(I + A) \\ A &{}= (Q - I)(Q + I)^{-1} \end{align}

The mappings may also be written with the order of the factors reversed (Courant & Hilbert 1989, Ch.VII, §7.2); however, A always commutes with (μI ± A)⁻¹, so the reordering does not affect the definition.

Examples

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 + \bold{i} x + \bold{j} y + \bold{k} z \,\!

(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).

Other matrices

We 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, we may extend the definition further with choices of adjoint other than transpose or conjugate transpose.

Formally, the definition only requires some invertibility, so we can substitute for Q any matrix M whose eigenvalues do not include −1. For example, we have

\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} .

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

Conformal map

Cayley transform of upper complex half-plane to unit disk

In complex analysis, the Cayley transform is a mapping of the complex plane to itself, given by

\operatorname{W} \colon z \mapsto \frac{z-\bold{i}}{z+\bold{i}} .

This is a Möbius transformation, and can be extended to an automorphism of the Riemann sphere (the complex plane augmented with a point at infinity).

Of particular note are the following facts:

W maps the upper half plane of C conformally onto the unit disc of C.
W maps the real line R injectively into the unit circle T (complex numbers of absolute value 1). The image of R is T with 1 removed.
W maps the upper imaginary axis i [0, ∞) bijectively onto the half-open interval [−1, +1).
W maps 0 to −1.
W maps the point at infinity to 1.
W maps −i to the point at infinity (so W has a pole at −i).
W maps −1 to i.
W maps both ¹⁄₂(−1 + √3)(−1 + i) and ¹⁄₂(1 + √3)(1 − i) to themselves.

Operator map

An infinite-dimensional version of an inner product space is a Hilbert space, and we can no longer speak of matrices. However, matrices are merely representations of linear operators, and these we still have. So, generalizing both the matrix mapping and the complex plane mapping, we may define a Cayley transform of operators.

\begin{align} U &{}= (A - \bold{i}I) (A + \bold{i}I)^{-1} \\ A &{}= \bold{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.

References

Cayley, Arthur (1846), "Sur quelques propriétés des déterminants gauches", Journal für die reine und angewandte Mathematik 32: 119–123, doi:10.1515/crll.1846.32.119, ISSN 0075-4102 ; reprinted as article 52 (pp. 332–336) in Cayley, Arthur (1889), The collected mathematical papers of Arthur Cayley, I (1841–1853), Cambridge University Press, pp. 332–336
Courant, Richard; Hilbert, David (1989), Methods of Mathematical Physics 1 (1st English ed.), New York: Wiley-Interscience, ISBN 978-0-471-50447-4
Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Baltimore: Johns Hopkins University Press, ISBN 978-0-8018-5414-9
Nikol’skii, N. K. (2001), "Cayley transform", Encyclopaedia of Mathematics, Springer-Verlag, ISBN 978-1-4020-0609-8 ; translated from the Russian Vinogradov, I. M., ed. (1977), Matematicheskaya Entsiklopediya, Moscow: Sovetskaya Entsiklopediya
Remmert, Reinhold (1991), Theory of Complex Functions, Graduate Texts in Mathematics, 122 of Graduate Texts in Mathematics (Readings in Mathematics), Translated by Robert B. Burckel (trans.), New York: Springer-Verlag, ISBN 978-0-387-97195-7 , translated by Robert B. Burckel from Remmert, Reinhold (1989), Funktionentheorie I (2nd ed.), Springer-Verlag, ISBN 978-3-540-51238-7
Rudin, Walter (1987), Real and Complex Analysis (3rd ed.), McGraw-Hill, ISBN 978-0-07-100276-9

External links

Cayley's parameterization of orthogonal matrices at PlanetMath.org.