Editing Rotation matrix (section)

===Relationship with complex plane===

The matrices of the shape
<math display=block>\begin{bmatrix} x & -y \\ y & x \end{bmatrix}</math> form a [[ring (mathematics)|ring]], since their set is closed under addition and multiplication.
Since
<math display=block>\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}^2 \ =\  \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \ = -I</math>
(where <math display="inline">I</math> is the [[identity matrix]]), the map
:<math>\begin{bmatrix} x & -y \\ y & x \end{bmatrix} = x\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + y \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \mapsto x+iy</math>
(where <math>i = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}</math>) is a [[ring isomorphism]] from this ring to the [[field (mathematics)|field]] of the [[complex number]]s {{tmath|\C}} (incidentally, this shows that this ring is a field). Under this isomorphism, the rotation matrices correspond to the [[circle]] of the [[unit complex number]]s, the complex numbers of modulus {{math|1}}.

If one identifies <math>\mathbb R^2</math> with <math>\mathbb C</math> through the [[linear isomorphism]] <math>(a,b)\mapsto a+ib</math>, where <math>(a,b) \in \mathbb R^2</math> and <math>a+ib \in \mathbb C</math>, the action of a matrix <math>\begin{bmatrix} x & -y \\ y & x \end{bmatrix}</math> on a vector <math>(a,b)</math> corresponds to multiplication on the complex number <math>a+ib</math> by {{math|''x'' + ''iy''}}, and a rotation correspond to multiplication by a complex number of modulus {{math|1}}.

As every 2-dimensional rotation matrix can be written
:<math>\begin{pmatrix}\cos t&-\sin t\\ \sin t&\cos t\end{pmatrix},</math>
the above correspondence associates such a matrix with the complex number
:<math>e^{it} = \cos t + i\sin t = \cos t \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} + \sin t \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}</math>
where the first equality is [[Euler's formula]], the matrix <math>I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}</math>corresponds to 1, and the matrix <math>\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}</math> corresponds to the [[imaginary unit]] <math display="inline">i</math>.