- Cayley transform
mathematics, the Cayley transform, named after Arthur Cayley, has a cluster of related meanings. As originally described by Harvtxt|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 Harv|Rudin|1987 in which the image of the upper complex half-plane is the unit disk Harv|Remmert|1991|pp=82ff, 275. And in the theory of Hilbert spaces, the Cayley transform is a mapping between linear operators Harv|Nikol’skii|2001.
orthogonal matrix, "Q" (so that "Q"T"Q" = "I"). 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
is a skew-symmetrix 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 Harv|Golub|Van Loan|1996, requiring different mappings in each direction (and dropping the superscript notation):
The mappings may also be written with the order of the factors reversed Harv|Courant|Hilbert|1989|loc=Ch.VII, §7.2; however, "A" always commutes with (μ"I" ± "A")−1, so the reordering does not affect the definition.
In the 2×2 case, we have: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:
where "K" = "w"2 + "x"2 + "y"2 + "z"2, and where "w" = 1. This we recognize as the rotation matrix corresponding to
(by a formula Cayley had published the year before), except scaled so that "w" = 1 instead of the usual scaling so that "w"2 + "x"2 + "y"2 + "z"2 = 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").
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 productwith the standard complex inner product. In fact, we may extend the definition further with choices of adjointother 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:We remark that "A" is skew-symmetric (respectively, skew-Hermitian) if and only if "Q" is orthogonal (respectively, unitary) with no eigenvalue −1.
complex analysis, the Cayley transform is a mapping of the complex planeto itself, given by
This is a
linear fractional transformation, and can be extended to an automorphismof the Riemann sphere(the complex planeaugmented 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 value1). 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.
An infinite-dimensional version of an
inner product spaceis 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.:Here the domain of "U", dom "U", is ("A"+i"I") dom "A". See self-adjoint operator for further details.
Extensions of symmetric operators
title=Sur quelques propriétés des déterminants gauches
journal=Journal für die Reine und Angewandte Mathematik (
issn=0075-4102; reprinted as article 52 (pp. 332–336) in Citation
title=The collected mathematical papers of Arthur Cayley
Cambridge University Press
title=Methods of Mathematical Physics
author1-link=Gene H. Golub
author2-link=Charles F. Van Loan
publisher=Johns Hopkins University Press
contribution= [http://eom.springer.de/C/c021100.htm Cayley transform]
Encyclopaedia of Mathematics
isbn=978-1-4020-0609-8 ; translated from the Russian Citation
editor-link=Ivan Matveyevich Vinogradov
translator=Robert B. Burckel (trans.)
title=Theory of Complex Functions
series=Graduate Texts in Mathematics
volume=122 of "Graduate Texts in Mathematics" ("Readings in Mathematics")
isbn=978-0-387-97195-7, translated by Robert B. Burckel from Citation
Grundwissen Mathematik 5
title=Real and Complex Analysis
title=Cayley's parameterization of orthogonal matrices
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