Editing Rotation matrix (section)

====Determining the axis====
[[File:Rotation decomposition.png|thumb|A rotation {{mvar|R}} around axis {{math|'''u'''}} can be decomposed using 3 endomorphisms {{math|'''P'''}}, {{math|('''I''' − '''P''')}}, and {{math|'''Q'''}} (click to enlarge).]]

Given a {{nowrap|3 × 3}} rotation matrix {{mvar|R}}, a vector {{math|'''u'''}} parallel to the rotation axis must satisfy
:<math>R\mathbf{u} = \mathbf{u},</math>
since the rotation of {{math|'''u'''}} around the rotation axis must result in {{math|'''u'''}}. The equation above may be solved for {{math|'''u'''}} which is  unique up to a scalar factor unless {{mvar|R}} is the [[identity matrix]] {{mvar|I}}.

Further, the equation may be rewritten
:<math>R\mathbf{u} = I \mathbf{u}  \implies  \left(R - I\right) \mathbf{u} = 0,</math>
which shows that {{math|'''u'''}} lies in the [[null space]] of {{math|''R'' − ''I''}}.

Viewed in another way, {{math|'''u'''}} is an [[eigenvector]] of {{mvar|R}} corresponding to the [[eigenvalue]] {{math|''λ'' {{=}} 1}}. Every rotation matrix must have this eigenvalue, the other two eigenvalues being [[complex conjugate]]s of each other. It follows that a general rotation matrix in three dimensions has, up to a multiplicative constant, only one real eigenvector.

One way to determine the rotation axis is by showing that:

:<math>\begin{align}
0 &= R^\mathsf{T} 0 + 0 \\
&= R^\mathsf{T}\left(R - I\right) \mathbf{u} + \left(R - I\right) \mathbf{u} \\
&= \left(R^\mathsf{T}R - R^\mathsf{T} + R - I\right) \mathbf{u} \\
&= \left(I - R^\mathsf{T} + R - I\right) \mathbf{u} \\
&= \left(R - R^\mathsf{T}\right) \mathbf{u}
\end{align}</math>

Since {{math|(''R'' − ''R''<sup>T</sup>)}} is a [[skew-symmetric matrix]], we can choose {{math|'''u'''}} such that
:<math>[\mathbf u]_{\times} = \left(R - R^\mathsf{T}\right).</math>
The matrix–vector product becomes a [[cross product]] of a vector with itself, ensuring that the result is zero:

:<math>\left(R - R^\mathsf{T}\right) \mathbf{u} = [\mathbf u]_{\times} \mathbf{u} = \mathbf{u} \times \mathbf{u} = 0\,</math>

Therefore, if
:<math>R = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix},</math>
then
:<math>\mathbf{u} = \begin{bmatrix} h-f \\ c-g \\ d-b \\ \end{bmatrix}.</math>
The magnitude of {{math|'''u'''}} computed this way is {{math|{{norm|'''u'''}} {{=}} 2 sin ''θ''}}, where {{mvar|θ}} is the angle of rotation.

This '''does not work''' if {{mvar|R}} is symmetric.  Above, if {{math|''R'' − ''R''<sup>T</sup>}} is zero, then all subsequent steps are invalid.  In this case, the angle of rotation is 0° or 180° and any nonzero column of {{math|''I'' + ''R''}} is an eigenvector of {{mvar|R}} with eigenvalue 1 because {{math|''R''(''I'' + ''R'') {{=}} ''R'' + ''R''<sup>2</sup> {{=}} ''R'' + ''RR''<sup>T</sup> {{=}} ''I'' + ''R''}}.<ref>{{Cite journal |last1=Palais |first1=Bob |last2=Palais |first2=Richard |date=2007-12-20 |title=Euler's fixed point theorem: The axis of a rotation |journal=Journal of Fixed Point Theory and Applications |language=en |volume=2 |issue=2 |pages=215–220 |doi=10.1007/s11784-007-0042-5 |issn=1661-7738 |mr=2372984 |doi-access=free}}</ref>