Editing Rotation matrix (section)

== Properties ==
For any {{mvar|n}}-dimensional rotation matrix {{mvar|R}} acting on <math>\mathbb{R}^n,</math>
: <math> R^\mathsf{T} = R^{-1}</math> (The rotation is an [[orthogonal matrix]])
It follows that:
: <math> \det R = \pm 1</math>

A rotation is termed proper if {{math|det ''R'' {{=}} 1}}, and [[improper rotation|improper]] (or a roto-reflection) if {{math|det ''R'' {{=}} –1}}. For even dimensions {{math|''n'' {{=}} 2''k''}}, the {{mvar|n}} [[eigenvalues]] {{mvar|λ}} of a proper rotation occur as pairs of [[complex conjugate]]s which are roots of unity: {{math|''λ'' {{=}} ''e''<sup>±''iθ<sub>j</sub>''</sup>}} for {{math|''j'' {{=}} 1, ..., ''k''}}, which is real only for {{math|''λ'' {{=}} ±1}}. Therefore, there may be no vectors fixed by the rotation ({{math|''λ'' {{=}} 1}}), and thus no axis of rotation. Any fixed eigenvectors occur in pairs, and the axis of rotation is an even-dimensional subspace.

For odd dimensions {{math|''n'' {{=}} 2''k'' + 1}}, a proper rotation {{mvar|R}} will have an odd number of eigenvalues, with at least one {{math|''λ'' {{=}} 1}} and the axis of rotation will be an odd dimensional subspace. Proof:
:<math>\begin{align}
  \det\left(R - I\right)
     &= \det\left(R^\mathsf{T}\right) \det\left(R - I\right)
      = \det\left(R^\mathsf{T}R - R^\mathsf{T}\right)
      = \det\left(I - R^\mathsf{T}\right) \\
     &= \det(I - R)
      = \left(-1\right)^n \det\left(R - I\right)
      = -\det\left(R - I\right).
\end{align}</math>

Here {{mvar|I}} is the identity matrix, and we use {{math|det(''R''<sup>T</sup>) {{=}} det(''R'') {{=}} 1}}, as well as {{math|(−1)<sup>''n''</sup> {{=}} −1}} since {{mvar|n}} is odd. Therefore, {{math|det(''R'' – ''I'') {{=}} 0}}, meaning there is a nonzero vector {{math|'''v'''}} with {{math|(''R – I'')'''v''' {{=}} 0}}, that is {{math|''R'''''v''' {{=}} '''v'''}}, a fixed eigenvector. There may also be pairs of fixed eigenvectors in the even-dimensional subspace orthogonal to {{math|'''v'''}}, so the total dimension of fixed eigenvectors is odd.

For example, in [[SO(2)|2-space]] {{math|''n'' {{=}} 2}}, a rotation by angle {{mvar|θ}} has eigenvalues {{math|''λ'' {{=}} ''e<sup>iθ</sup>''}} and {{math|''λ'' {{=}} ''e''<sup>−''iθ''</sup>}}, so there is no axis of rotation except when {{math|''θ'' {{=}} 0}}, the case of the null rotation. In [[SO(3)|3-space]] {{math|''n'' {{=}} 3}}, the axis of a non-null proper rotation is always a unique line, and a rotation around this axis by angle {{mvar|θ}} has eigenvalues {{math|''λ'' {{=}} 1, ''e<sup>iθ</sup>'', ''e''<sup>−''iθ''</sup>}}. In [[SO(4)|4-space]] {{math|''n'' {{=}} 4}}, the four eigenvalues are of the form {{math|''e''<sup>±''iθ''</sup>, ''e''<sup>±''iφ''</sup>}}. The null rotation has {{math|''θ'' {{=}} ''φ'' {{=}} 0}}. The case of {{math|''θ'' {{=}} 0, ''φ'' ≠ 0}} is called a ''simple rotation'', with two unit eigenvalues forming an ''axis plane'', and a two-dimensional rotation orthogonal to the axis plane. Otherwise, there is no axis plane. The case of {{math|''θ'' {{=}} ''φ''}} is called an ''isoclinic rotation'', having eigenvalues {{math|''e''<sup>±''iθ''</sup>}} repeated twice, so every vector is rotated through an angle {{mvar|θ}}.

The trace of a rotation matrix is equal to the sum of its eigenvalues. For {{math|''n'' {{=}} 2}}, a rotation by angle {{mvar|θ}} has trace {{math|2 cos ''θ''}}. For {{math|''n'' {{=}} 3}}, a rotation around any axis by angle {{mvar|θ}} has trace {{math|1 + 2 cos ''θ''}}. For {{math|''n'' {{=}} 4}}, and the trace is {{math|2(cos ''θ'' + cos ''φ'')}}, which becomes {{math|4 cos ''θ''}} for an isoclinic rotation.