Inmathematics, theCayley transform, named afterArthur Cayley, is any of a cluster of related things. As originally described byCayley (1846), the Cayley transform is a mapping betweenskew-symmetric matrices andspecial orthogonal matrices. The transform is ahomography used inreal analysis,complex analysis, andquaternionic analysis. In the theory ofHilbert spaces, the Cayley transform is a mapping betweenlinear operators (Nikolski 1988).

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

As a realhomography, points are described withprojective coordinates, and the mapping is

On theupper half of thecomplex plane, the Cayley transform is:^[1][2]

${\displaystyle f(z)=\frac{z-i}{z+i}.}$

Since${\displaystyle \{\mathrm{\infty },1,-1\}}$ is mapped to${\displaystyle \{1,-i,i\}}$, andMöbius transformations permute thegeneralised circles in thecomplex plane,${\displaystyle f}$ maps the real line to theunit circle. Furthermore, since${\displaystyle f}$ is ahomeomorphism and${\displaystyle i}$ is taken to 0 by${\displaystyle f}$, the upper half-plane is mapped to theunit disk.

In terms of themodels ofhyperbolic geometry, this Cayley transform relates thePoincaré half-plane model to thePoincaré disk model.

In electrical engineering the Cayley transform has been used to map areactance half-plane to theSmith chart used forimpedance matching of transmission lines.

In thefour-dimensional space ofquaternions${\displaystyle a+b\overrightarrow{i}+c\overrightarrow{j}+d\overrightarrow{k}}$, theversors

${\displaystyle u(\theta ,r)=\cos \theta +r\sin \theta }$ form the unit3-sphere.

Since quaternions are non-commutative, elements of itsprojective line have homogeneous coordinates written${\displaystyle U[a,b]}$ 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${\displaystyle \theta }$ is zero or${\displaystyle \pi /2}$, respectively.Evidently the transform takes${\displaystyle u\to 0\to -1}$ and takes${\displaystyle -u\to \mathrm{\infty }\to 1}$.

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

${\displaystyle f(u,1)=(1+u{)}^{-1}(1-u)=(1+u{)}^{\ast }(1-u)/|1+u{|}^2.}$

But${\displaystyle |1+u{|}^2=(1+u)(1+u^{\ast })=2+2\cos \theta ,\text{and}(1+u^{\ast })(1-u)=-2r\sin \theta .}$

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

In this form the Cayley transform has been described as a rational parametrization of rotation: Let${\displaystyle t=\tan \varphi /2}$ in the complex number identity^[3]

${\displaystyle e^{-i\phi }=\frac{1-ti}{1+ti}}$

where the right hand side is the transform of${\displaystyle ti}$ and the left hand side represents the rotation of the plane by negative${\displaystyle \varphi }$ radians.

Let${\displaystyle u^{\ast }=\cos \theta -r\sin \theta =u^{-1}.}$ Since

where the equivalence is in theprojective linear group over quaternions, theinverse of${\displaystyle f(u,1)}$ is

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

Amongn×nsquare matrices over thereals, withI theidentity matrix, letA be anyskew-symmetric matrix (so thatA^T = −A).

ThenI + A isinvertible, and the Cayley transform

${\displaystyle Q=(I-A)(I+A{)}^{-1}}$

produces anorthogonal matrix,Q (so thatQ^TQ =I). The matrix multiplication in the definition ofQ above is commutative, soQ can be alternatively defined as${\displaystyle Q=(I+A{)}^{-1}(I-A)}$. In fact,Q must have determinant +1, so is special orthogonal.

Conversely, letQ be any orthogonal matrix which does not have −1 as aneigenvalue; then

${\displaystyle A=(I-Q)(I+Q{)}^{-1}}$

is a skew-symmetric matrix. (See also:Involution.) The condition onQ automatically excludes matrices with determinant −1, but also excludes certain special orthogonal matrices.

However, any rotation (special orthogonal) matrixQ can be written as

${\displaystyle Q=((I-A)(I+A{)}^{-1}{)}^2}$

for some skew-symmetric matrixA; more generally any orthogonal matrixQ can be written as

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

for some skew-symmetric matrixA and some diagonal matrixE with ±1 as entries.^[4]

A slightly different form is also seen,^[5][6] requiring different mappings in each direction,

The mappings may also be written with the order of the factors reversed;^[7][8] however,A always commutes with (μI ± 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

whereK = w² + x² + y² + z², and wherew = 1. This we recognize as the rotation matrix corresponding toquaternion

${\displaystyle w+\mathbf{i}x+\mathbf{j}y+\mathbf{k}z}$

(by a formula Cayley had published the year before), except scaled so thatw = 1 instead of the usual scaling so thatw² + 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 allQ which aresymmetric (so thatQ^T =Q).

One can extend the mapping tocomplex matrices by substituting "unitary" for "orthogonal" and "skew-Hermitian" for "skew-symmetric", the difference being that the transpose (·^T) is replaced by theconjugate transpose (·^H). This is consistent with replacing the standard realinner product with the standard complex inner product. In fact, one may extend the definition further with choices ofadjoint other than transpose or conjugate transpose.

Formally, the definition only requires some invertibility, so one can substitute forQ any matrixM whose eigenvalues do not include −1. For example,

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

An infinite-dimensional version of aninner product space is aHilbert space, and one can no longer speak ofmatrices. However, matrices are merely representations oflinear 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.^[9]

Here the domain ofU, dom U, is (A+iI) dom A. Seeself-adjoint operator for further details.

• Bilinear transform
• Extensions of symmetric operators

• Sterling K. Berberian (1974)Lectures in Functional Analysis and Operator Theory,Graduate Texts in Mathematics #15, pages 278, 281, Springer-VerlagISBN 978-0-387-90081-0
• Cayley, Arthur (1846),"Sur quelques propriétés des déterminants gauches",Journal für die reine und angewandte Mathematik,32 (2):119–123,doi:10.1515/crll.1846.32.119,ISSN 0075-4102; reprinted as article 52 (pp. 332–336) inCayley, Arthur (1889),The collected mathematical papers of Arthur Cayley, vol. I (1841–1853),Cambridge University Press, pp. 332–336
• Lokenath Debnath & Piotr Mikusiński (1990)Introduction to Hilbert Spaces with Applications, page 213,Academic PressISBN 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
• Nikolski, Nikolai Kapitonovich (1988),"Cayley transform", inHazewinkel, Michiel (ed.),Encyclopaedia of Mathematics, vol. 2, Kluwer, p. 80,doi:10.1007/978-94-009-6000-8,ISBN 978-94-009-6002-2; translated from the RussianVinogradov, Ivan Matveevich, ed. (1977),Matematicheskaya Entsiklopediya, Moscow: Sovetskaya Entsiklopediya
• Henry Ricardo (2010)A Modern Introduction to Linear Algebra, page 504,CRC PressISBN 978-1-4398-0040-9 .
• Rudin, Walter (1991).Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY:McGraw-Hill Science/Engineering/Math.ISBN 978-0-07-054236-5.OCLC 21163277.

• Conformal mappings
• Transforms

• Articles with short description
• Short description matches Wikidata
• CS1: long volume value