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. The transform is a homography used in real analysis, complex analysis, and quaternionic analysis. In the theory of Hilbert spaces, the Cayley transform is a mapping between linear operators (Nikol’skii 2001).

Real homography

The Cayley transform is an automorphism of the real projective line that permutes 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 homogeneous coordinates, and the mapping is

(y,\ 1)=\left({\frac {x-1}{x+1}},\ 1\right)\sim (x-1,\ x+1)=(x,\ 1){\begin{pmatrix}1&1\\-1&1\end{pmatrix}}.

Complex homography

Cayley transform of upper complex half-plane to unit disk

In the complex projective plane the Cayley transform is:^[1]^[2]

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

Since {∞, 1, –1 } is mapped to {1, –i, i }, and Möbius transformations permute the generalised circles in the complex plane, f maps the real line to the unit circle. Furthermore, since f is continuous 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.

Quaternion homography

In the four-dimensional space of quaternions q = a + b i + c j + d k, the versors

u(\theta ,r)=\cos \theta +r\ \sin \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

f(u,q)=U(q,1){\begin{pmatrix}1&1\\-u&u\end{pmatrix}}=U(q-u,\ q+u)\sim U((q+u)^{-1}(q-u),\ 1).

The real and complex homographies described above are instances of the quaternion homography where θ is zero or π/2, respectively. Evidently the transform takes u → 0 → –1 and takes –u → ∞ → 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 $|1+u|^{2}=(1+u)(1+u^{*})=2+2\cos \theta ,\quad {\text{and}}\quad (1+u^{*})(1-u)=-2r\sin \theta .$

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

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

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

where the right hand side is the transform of t i and the left hand side represents the rotation of the plane by negative φ radians.

Inverse

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

{\begin{pmatrix}1&1\\-u&u\end{pmatrix}}\ {\begin{pmatrix}1&-u^{*}\\1&u^{*}\end{pmatrix}}\ =\ {\begin{pmatrix}2&0\\0&2\end{pmatrix}}\ \sim \ {\begin{pmatrix}1&0\\0&1\end{pmatrix}}\ ,

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

U(p,1)\ {\begin{pmatrix}1&-u^{*}\\1&u^{*}\end{pmatrix}}\ =\ U(p+1,\ (1-p)u^{*})\sim U(u(1-p)^{-1}(p+1),\ 1).

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 ℝ³.

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.

A slightly different form is also seen (Golub & Van Loan 1996), requiring different mappings in each direction:

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

The mappings may also be written with the order of the factors reversed;^[4]^[5] 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(xy-wz)&2(wy+xz)\\2(xy+wz)&w^{2}-x^{2}+y^{2}-z^{2}&2(yz-wx)\\2(xz-wy)&2(wx+yz)&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+{\mathbf {i}}x+{\mathbf {j}}y+{\mathbf {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.

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

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

References

↑ 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
↑ See Tangent half-angle formula
↑ Courant, Richard; Hilbert, David (1989), Methods of Mathematical Physics, 1 (1st English ed.), New York: Wiley-Interscience, pp. 536, 7, ISBN 978-0-471-50447-4 Ch.VII, §7.2
↑ Howard Eves (1966) Elementary Matrix Theory, § 5.4A Cayley’s Construction of Real Orthogonal Matrices, pages 365–7, Allyn & Bacon

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
Cayley, Arthur (1846), "Sur quelques propriétés des déterminants gauches", Journal für die reine und angewandte Mathematik, 32: 119–123, ISSN 0075-4102, doi:10.1515/crll.1846.32.119 ; 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
Lokenath Debnath & Piotr Mikusiński (1990) Introduction to Hilbert Spaces with Applications, page 213, Academic Press ISBN 0-12-208435-7
Golub, Gene H.; Van Loan, Charles F. (1996), Matrix Computations (3rd ed.), Baltimore: Johns Hopkins University Press, ISBN 978-0-8018-5414-9
Gilbert Helmberg (1969) Introduction to Spectral Theory in Hilbert Space, page 288, § 38: The Cayley Transform, Applied Mathematics and Mechanics #6, North Holland
Henry Ricardo (2010) A Modern Introduction to Linear Algebra, page 504, CRC Press ISBN 978-1-4398-0040-9 .

External links

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

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