Editing Cayley transform (section)

== Matrix map ==
Among ''n''×''n'' [[square matrix|square matrices]] over the [[real number|reals]], with ''I'' the [[identity matrix]], let ''A'' be any [[skew-symmetric matrix]] (so that ''A''<sup>T</sup>&nbsp;= −''A''). 

Then ''I''&nbsp;+&nbsp;''A'' is [[invertible matrix|invertible]], and the Cayley transform
:<math> Q = (I - A)(I + A)^{-1} \,\!</math>
produces an [[orthogonal matrix]], ''Q'' (so that ''Q''<sup>T</sup>''Q''&nbsp;= ''I''). The matrix multiplication in the definition of ''Q'' above is commutative, so ''Q'' can be alternatively defined as <math> Q = (I + A)^{-1}(I - A)</math>. 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
:<math> A = (I - Q)(I + Q)^{-1} \,\!</math>
is a skew-symmetric matrix.  (See also: [[Involution (mathematics)|Involution]].) The condition on ''Q'' automatically excludes matrices with determinant −1, but also excludes certain special orthogonal matrices.

However, any rotation (special orthogonal) matrix ''Q'' can be written as

:<math>Q = \bigl((I - A)(I + A)^{-1}\bigr)^2</math>

for some skew-symmetric matrix ''A''; more generally any orthogonal matrix ''Q'' can be written as

:<math>Q = E(I - A)(I + A)^{-1}</math>

for some skew-symmetric matrix ''A'' and some diagonal matrix ''E'' with ±1 as entries.<ref>{{cite arXiv |last=Gallier |first=Jean |author-link=Jean Gallier |title=Remarks on the Cayley Representation of Orthogonal Matrices and on Perturbing the Diagonal of a Matrix to Make it Invertible | eprint=math/0606320 |year=2006}}{{pb}}
As described by Gallier, the first of these results is a sharpened variant of {{cite book |last=Weyl |first=Hermann |author-link=Hermann Weyl |year=1946 |title=The Classical Groups |edition=2nd |publisher=Princeton University Press |at=Lemma 2.10.D, p. 60 }}{{pb}}
The second appeared as an exercise in {{cite book |last=Bellman |first=Richard |title=Introduction to Matrix Analysis |publisher=SIAM Publications |year=1960 |at=§6.4 exercise 11, p. 91–92 }}
</ref>

A slightly different form is also seen,<ref>{{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]] | year=1996 | isbn=978-0-8018-5414-9}}</ref><ref>F. Chong (1971) "A Geometric Note on the Cayley Transform", pages 84,5 in ''A Spectrum of Mathematics: Essays Presented to H. G. Forder'', [[John C. Butcher]] editor, [[Auckland University Press]]</ref> requiring different mappings in each direction,
:<math>\begin{align}
 Q &= (I - A)^{-1}(I + A), \\[5mu]
 A &= (Q - I)(Q + I)^{-1}.
\end{align}</math>

The mappings may also be written with the order of the factors reversed;<ref>{{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  | year=1989  | pages=536, 7  | place=New York  | isbn=978-0-471-50447-4}} Ch.VII,&nbsp;§7.2</ref><ref>[[Howard Eves]] (1966) ''Elementary Matrix Theory'', § 5.4A Cayley’s Construction of Real Orthogonal Matrices, pages 365–7, [[Allyn & Bacon]]</ref> however, ''A'' always commutes with (μ''I''&nbsp;±&nbsp;''A'')<sup>−1</sup>, so the reordering does not affect the definition.

=== Examples ===
In the 2×2 case, we have
:<math>
\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} .
</math>
The 180° [[rotation matrix]], −''I'', is excluded, though it is the limit as tan&nbsp;<sup>θ</sup>⁄<sub>2</sub> goes to infinity.

In the 3×3 case, we have
:<math>
\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 (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^2
\end{bmatrix} ,
</math>

where ''K''&nbsp;=&nbsp;''w''<sup>2</sup>&nbsp;+&nbsp;''x''<sup>2</sup>&nbsp;+&nbsp;''y''<sup>2</sup>&nbsp;+&nbsp;''z''<sup>2</sup>, and where ''w''&nbsp;=&nbsp;1. This we recognize as the rotation matrix corresponding to [[quaternion]]

:<math> w + \mathbf{i} x + \mathbf{j} y + \mathbf{k} z \,\!</math>

(by a formula Cayley had published the year before), except scaled so that ''w''&nbsp;= 1 instead of the usual scaling so that ''w''<sup>2</sup>&nbsp;+&nbsp;''x''<sup>2</sup>&nbsp;+&nbsp;''y''<sup>2</sup>&nbsp;+&nbsp;''z''<sup>2</sup>&nbsp;=&nbsp;1. Thus vector (''x'',''y'',''z'') is the unit axis of rotation scaled by tan&nbsp;<sup>θ</sup>⁄<sub>2</sub>. Again excluded are 180° rotations, which in this case are all ''Q'' which are [[symmetric matrix|symmetric]] (so that ''Q''<sup>T</sup>&nbsp;= ''Q'').

=== Other matrices ===
One can extend the mapping to [[complex number|complex]] matrices by substituting "[[unitary matrix|unitary]]" for "orthogonal" and "[[skew-Hermitian matrix|skew-Hermitian]]" for "skew-symmetric", the difference being that the transpose (·<sup>T</sup>) is replaced by the [[conjugate transpose]] (·<sup>H</sup>). This is consistent with replacing the standard real [[inner product]] with the standard complex inner product. In fact, one may extend the definition further with choices of [[Hermitian adjoint|adjoint]] 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,
:<math>
\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} .
</math>
Note that ''A'' is skew-symmetric (respectively, skew-Hermitian) if and only if ''Q'' is orthogonal (respectively, unitary) with no eigenvalue −1.