Editing Euler's rotation theorem (section)

==Equivalence of an orthogonal matrix to a rotation matrix==
Two matrices (representing linear maps) are said to be equivalent if there is a [[change of basis]] that makes one equal to the other. A proper [[orthogonal matrix]] is always equivalent (in this sense) to either the following matrix or to its vertical reflection:
:<math>
\mathbf{R} \sim
\begin{pmatrix}
\cos\phi & -\sin\phi & 0 \\
\sin\phi & \cos\phi & 0 \\
0 & 0 & 1\\
\end{pmatrix}, \qquad 0\le \phi \le 2\pi.
</math>

Then, any orthogonal matrix is either a rotation or an [[improper rotation]]. A general orthogonal matrix has only one real eigenvalue, either +1 or&nbsp;−1. When it is +1 the matrix is a rotation. When −1, the matrix is an improper rotation.

If {{math|'''R'''}} has more than one invariant vector then {{math|''φ'' {{=}} 0}} and {{math|'''R''' {{=}} '''I'''}}. ''Any'' vector is an invariant vector of {{math|'''I'''}}.

===Excursion into matrix theory===
In order to prove the previous equation some facts from matrix theory must be recalled.

An {{math|''m'' × ''m''}} matrix {{math|'''A'''}} has {{math|''m''}} orthogonal eigenvectors if and only if {{math|'''A'''}} is [[normal matrix|normal]], that is, if {{math|'''A'''<sup>†</sup>'''A''' {{=}} '''AA'''<sup>†</sup>}}.{{efn|The dagger symbol {{math|†}} stands for [[complex conjugation]] followed by transposition. For real matrices complex conjugation does nothing and daggering a real matrix is the same as transposing it.}} This result is equivalent to stating that normal matrices can be brought to diagonal form by a unitary similarity transformation:
:<math>
\mathbf{A}\mathbf{U} = \mathbf{U}\; \operatorname{diag}(\alpha_1,\ldots,\alpha_m)\quad \Longleftrightarrow\quad
\mathbf{U}^\dagger \mathbf{A}\mathbf{U} = \operatorname{diag}(\alpha_1,\ldots,\alpha_m),
</math>
and {{math|'''U'''}} is unitary, that is,
:<math>
\mathbf{U}^\dagger = \mathbf{U}^{-1}.
</math>
The eigenvalues {{math|''α''<sub>1</sub>, ..., ''α<sub>m</sub>''}} are roots of the characteristic equation. If the matrix {{math|'''A'''}} happens to be unitary (and note that unitary matrices are normal), then
:<math>
\left(\mathbf{U}^\dagger\mathbf{A} \mathbf{U}\right)^\dagger = \operatorname{diag}\left(\alpha^*_1,\ldots,\alpha^*_m\right) =
\mathbf{U}^\dagger\mathbf{A}^{-1} \mathbf{U} = \operatorname{diag}\left(\frac{1}{\alpha_1},\ldots,\frac{1}{\alpha_m}\right)
</math>
and it follows that the eigenvalues of a unitary matrix are on the unit circle in the complex plane:
:<math>
\alpha^*_k = \frac{1}{\alpha_k} \quad\Longleftrightarrow\quad \alpha^*_k\alpha_k = \left|\alpha_k\right|^2 = 1,\qquad k=1,\ldots,m.
</math>
Also an orthogonal (real unitary) matrix has eigenvalues on the unit circle in the complex plane. Moreover, since its characteristic equation (an {{mvar|m}}th order polynomial in {{mvar|λ}}) has real coefficients, it follows that its roots appear in complex conjugate pairs, that is, if {{mvar|α}} is a root then so is {{math|''α''<sup>∗</sup>}}. There are 3 roots, thus at least one of them must be purely real (+1 or −1).

After recollection of these general facts from matrix theory, we return to the rotation matrix {{math|'''R'''}}. It follows from its realness and orthogonality that we can find a {{math|'''U'''}} such that:
:<math>
 \mathbf{R} \mathbf{U} = \mathbf{U}
\begin{pmatrix}
e^{i\phi} & 0 & 0 \\
0 & e^{-i\phi} & 0 \\
0 & 0 & \pm 1 \\
\end{pmatrix}
</math>
If a matrix {{math|'''U'''}} can be found that gives the above form, and there is only one purely real component and it is −1, then we define <math>\mathbf{R}</math> to be an improper rotation. Let us only consider the case, then, of matrices R that are proper rotations (the third eigenvalue is just 1). The third column of the {{nowrap|3 × 3}} matrix {{math|'''U'''}} will then be equal to the invariant vector {{math|'''n'''}}. Writing {{math|'''u'''<sub>1</sub>}} and {{math|'''u'''<sub>2</sub>}} for the first two columns of {{math|'''U'''}}, this equation gives
:<math>
 \mathbf{R}\mathbf{u}_1 = e^{i\phi}\, \mathbf{u}_1 \quad\hbox{and}\quad \mathbf{R}\mathbf{u}_2 = e^{-i\phi}\, \mathbf{u}_2.
</math>
If {{math|'''u'''<sub>1</sub>}} has eigenvalue 1, then {{math|''φ'' {{=}} 0}} and {{math|'''u'''<sub>2</sub>}} has also eigenvalue 1, which implies that in that case {{math|'''R''' {{=}} '''I'''}}. In general, however, as
<math> (\mathbf{R}-e^{i\phi}\mathbf{I})\mathbf{u}_1 = 0 </math> implies that also <math> (\mathbf{R}-e^{-i\phi}\mathbf{I})\mathbf{u}^*_1 = 0 </math> holds, so <math> \mathbf{u}_2 = \mathbf{u}^*_1 </math> can be chosen for <math> \mathbf{u}_2 </math>. Similarly, <math> (\mathbf{R}-\mathbf{I})\mathbf{u}_3 = 0 </math> can result in  a <math> \mathbf{u}_3 </math>  with real entries only, for a proper rotation matrix <math>\mathbf{R}</math>.  
Finally, the matrix equation is transformed by means of a unitary matrix,
:<math>
 \mathbf{R} \mathbf{U}
\begin{pmatrix}
\frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\
\frac{1}{\sqrt{2}} & \frac{-i}{\sqrt{2}} & 0 \\
0 & 0 & 1\\
\end{pmatrix}
= \mathbf{U}
\underbrace{
\begin{pmatrix}
\frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\
\frac{1}{\sqrt{2}} & \frac{-i}{\sqrt{2}} & 0 \\
0 & 0 & 1\\
\end{pmatrix}
\begin{pmatrix}
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 \\
\frac{-i}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\
0 & 0 & 1\\
\end{pmatrix}
}_{=\;\mathbf{I}}
\begin{pmatrix}
e^{i\phi} & 0 & 0 \\
0 & e^{-i\phi} & 0 \\
0 & 0 & 1 \\
\end{pmatrix}
\begin{pmatrix}
\frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\
\frac{1}{\sqrt{2}} & \frac{-i}{\sqrt{2}} & 0 \\
0 & 0 & 1\\
\end{pmatrix}
</math>
which gives
:<math>
\mathbf{U'}^\dagger \mathbf{R} \mathbf{U'} = \begin{pmatrix}
\cos\phi & -\sin\phi & 0 \\
\sin\phi & \cos\phi & 0 \\
0 & 0 & 1\\
\end{pmatrix}
\quad\text{ with }\quad \mathbf{U'}
= \mathbf{U}
\begin{pmatrix}
\frac{1}{\sqrt{2}} & \frac{i}{\sqrt{2}} & 0 \\
\frac{1}{\sqrt{2}} & \frac{-i}{\sqrt{2}} & 0 \\
0 & 0 & 1\\
\end{pmatrix} .
</math>
The columns of {{math|'''U'''′}} are orthonormal as it is a unitary matrix with real-valued entries only, due to its definition above, that <math> \mathbf{u}_1 </math> is the complex conjugate of  <math> \mathbf{u}_2 </math> and that <math> \mathbf{u}_3 </math> is a vector with real-valued components. The third column is still <math> \mathbf{u}_3 =</math> {{math|'''n'''}}, the other two columns of {{math|'''U'''′}} are perpendicular to {{math|'''n'''}}. We can now see how our definition of improper rotation corresponds with the geometric interpretation: an improper rotation is a rotation around an axis (here, the axis corresponding to the third coordinate) and a reflection on a plane perpendicular to that axis. If we only restrict ourselves to matrices with determinant 1, we can thus see that they must be proper rotations. This result implies that any orthogonal matrix {{math|'''R'''}} corresponding to a proper rotation is equivalent to a rotation over an angle {{mvar|φ}} around an axis {{math|'''n'''}}.