Cayley transform

Cayley transform

In 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.

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"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

: A = (I - Q)(I + Q)^{-1} ,!

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):

:egin{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 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.

Examples

In the 2×2 case, we have:egin{bmatrix} 0 & an frac{ heta}{2} \ - an frac{ heta}{2} & 0 end{bmatrix}lrarregin{bmatrix} cos heta & -sin heta \ sin heta & cos heta 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:egin{bmatrix} 0 & z & -y \ -z & 0 & x \ y & -x & 0 end{bmatrix}lrarrfrac{1}{K}egin{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^2end{bmatrix} ,

where "K" = "w"2 + "x"2 + "y"2 + "z"2, and where "w" = 1. This we recognize as the rotation matrix corresponding to quaternion

: w + old{i} x + old{j} y + old{k} z ,!

(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").

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:egin{bmatrix} 0 & -a & ab - c \ 0 & 0 & -b \ 0 & 0 & 0 end{bmatrix}lrarregin{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

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

: operatorname{W} colon z mapsto frac{z-old{i{z+old{i .

This is a linear fractional 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.:egin{align} U &{}= (A - old{i}I) (A + old{i}I)^{-1} \ A &{}= old{i}(I + U) (I - U)^{-1}end{align}Here the domain of "U", dom "U", is ("A"+i"I") dom "A". See self-adjoint operator for further details.

See also

* Bilinear transform

* Extensions of symmetric operators

References

* 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 (Crelle's Journal),
volume=32
pages=119–123
url=http://dz-srv1.sub.uni-goettingen.de/sub/digbib/loader?ht=VIEW&did=D268141
issn=0075-4102; reprinted as article 52 (pp. 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=
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
* 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
date=1989
place=New York
isbn=978-0-471-50447-4
* 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
date=1996
place=Baltimore
isbn=978-0-8018-5414-9
* Citation
last=Nikol’skii
first=N. K.
contribution= [http://eom.springer.de/C/c021100.htm Cayley transform]
title=Encyclopaedia of Mathematics
year=2001
publisher=Springer-Verlag
isbn=978-1-4020-0609-8 ; translated from the Russian Citation
editor-last=Vinogradov
editor-first=I. M.
editor-link=Ivan Matveyevich Vinogradov
title=Matematicheskaya Entsiklopediya
place=Moscow
publisher=Sovetskaya Entsiklopediya
year=1977
* Citation
last=Remmert
first=Reinhold
author-link=Reinhold Remmert
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")
year=1991
publisher=Springer-Verlag
place=New York
isbn=978-0-387-97195-7, translated by Robert B. Burckel from Citation
last=Remmert
first=Reinhold
author-link=Reinhold Remmert
title=Funktionentheorie I
edition=2nd
year=1989
Grundwissen Mathematik 5
publisher=Springer-Verlag
isbn=978-3-540-51238-7
* Citation
last=Rudin
first=Walter
author-link=Walter Rudin
title=Real and Complex Analysis
edition=3rd
publisher=McGraw-Hill
year=1987
isbn=978-0-07-100276-9

External links

* PlanetMath
urlname=CayleyTransform
title=Cayley's parameterization of orthogonal matrices
id=6535


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