In
mathematics, the Cayley transform, named after
Arthur Cayley, is any of a cluster of related things. As originally described by , the Cayley transform is a mapping between
skew-symmetric matrices
In mathematics, particularly in linear algebra, a skew-symmetric (or antisymmetric or antimetric) matrix is a square matrix whose transpose equals its negative. That is, it satisfies the condition
In terms of the entries of the matrix, if a_ ...
and
special orthogonal matrices. The transform is a
homography used in
real analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include conv ...
,
complex analysis, and
quaternionic analysis. In the theory of
Hilbert spaces, the Cayley transform is a mapping between
linear operators .
Real homography
The Cayley transform is an automorphism of the
real projective line
In geometry, a real projective line is a projective line over the real numbers. It is an extension of the usual concept of a line that has been historically introduced to solve a problem set by visual perspective: two parallel lines do not inters ...
that permutes the elements of in sequence. For example, it maps the
positive real numbers
In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...
to the interval
��1, 1 Thus the Cayley transform is used to adapt
Legendre polynomials
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applica ...
for use with functions on the positive real numbers with
Legendre rational functions
In mathematics the Legendre rational functions are a sequence of orthogonal functions on , ∞). They are obtained by composing the Cayley transform with Legendre polynomials">Cayley_transform.html" ;"title=", ∞). They are obta ...
.
As a real
homography, points are described with
projective coordinates, and the mapping is
:
Complex homography
On the
Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers ...
, the Cayley transform is:
:
Since is mapped to , and
Möbius transformations permute the
generalised circle
In geometry, a generalized circle, also referred to as a "cline" or "circline", is a straight line or a circle. The concept is mainly used in inversive geometry, because straight lines and circles have very similar properties in that geometry and ...
s in the
complex plane, ''f'' maps the real line to the
unit circle
In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
. 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
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For any given line ''R'' and point ''P ...
, 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
The Smith chart, invented by Phillip H. Smith (1905–1987) and independently by Mizuhashi Tosaku, is a graphical calculator or nomogram designed for electrical and electronics engineers specializing in radio frequency (RF) engineering to assis ...
used for
impedance matching
In electronics, impedance matching is the practice of designing or adjusting the input impedance or output impedance of an electrical device for a desired value. Often, the desired value is selected to maximize power transfer or minimize si ...
of transmission lines.
Quaternion homography
In the
four-dimensional space of
quaternions ''q'' = ''a'' + ''b'' i + ''c'' j + ''d'' k, the
versor
In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Will ...
s
:
form the unit
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensio ...
.
Since quaternions are non-commutative, elements of its
projective line
In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
have homogeneous coordinates written 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 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:
:
But
Thus
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
:
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
Since
:
where the equivalence is in the
projective linear group
In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associate ...
over quaternions, the
inverse of ''f''(''u'', 1) is
:
Since homographies are
bijections,
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 ℝ
3.
Matrix map
Among ''n''×''n''
square matrices
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
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
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers.
Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
, and the Cayley transform
:
produces an
orthogonal matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose of and is the identity m ...
, ''Q'' (so that ''Q''
T''Q'' = ''I''). The matrix multiplication in the definition of ''Q'' above is commutative, so ''Q'' can be alternatively defined as
. 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
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
; then
:
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, requiring different mappings in each direction,
:
The mappings may also be written with the order of the factors reversed;
Howard Eves
Howard Whitley Eves (10 January 1911, New Jersey – 6 June 2004) was an American mathematician, known for his work in geometry and the history of mathematics.
Eves received his B.S. from the University of Virginia, an M.A. from Harvard Universi ...
(1966) ''Elementary Matrix Theory'', § 5.4A Cayley’s Construction of Real Orthogonal Matrices, pages 365–7, Allyn & Bacon
Allyn & Bacon, founded in 1868, is a higher education textbook publisher in the areas of education, humanities and social sciences. It is an imprint of Pearson Education, the world's largest education publishing and technology company which is par ...
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
:
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
quaternion
:
(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
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
(so that ''Q''
T = ''Q'').
Other matrices
One can extend the mapping to
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
matrices by substituting "
unitary" for "orthogonal" and "
skew-Hermitian
__NOTOC__
In linear algebra, a square matrix with Complex number, complex entries is said to be skew-Hermitian or anti-Hermitian if its conjugate transpose is the negative of the original matrix. That is, the matrix A is skew-Hermitian if it satisf ...
" for "skew-symmetric", the difference being that the transpose (·
T) is replaced by the
conjugate transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...
(·
H). This is consistent with replacing the standard real
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
with the standard complex inner product. In fact, one may extend the definition further with choices of
adjoint
In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type
:(''Ax'', ''y'') = (''x'', ''By'').
Specifically, adjoin ...
other than transpose or conjugate transpose.
Formally, the definition only requires some invertibility, so one can substitute for ''Q'' any matrix ''M'' whose eigenvalues do not include −1. For example,
:
Note 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
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
is a
Hilbert space, and one can no longer speak of
matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
. However, matrices are merely representations of
linear 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.
:
Here the domain of ''U'', dom ''U'', is (''A''+i''I'') dom ''A''. See
self-adjoint operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to its ...
for further details.
See also
*
Bilinear transform
*
Extensions of symmetric operators
References
* Sterling K. Berberian (1974) ''Lectures in Functional Analysis and Operator Theory'',
Graduate Texts in Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard ...
#15, pages 278, 281, Springer-Verlag
* ; reprinted as article 52 (pp. 332–336) in
* Lokenath Debnath & Piotr Mikusiński (1990) ''Introduction to Hilbert Spaces with Applications'', page 213,
Academic Press
Academic Press (AP) is an academic book publisher founded in 1941. It was acquired by Harcourt, Brace & World in 1969. Reed Elsevier bought Harcourt in 2000, and Academic Press is now an imprint of Elsevier.
Academic Press publishes referen ...
* 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 .
External links
*
* ; 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
Conformal mappings
Transforms
ru:Преобразование Мёбиуса#Примеры