Editing Euler's rotation theorem (section)

==Equivalence classes==
The [[Trace (linear algebra)|trace]] (sum of diagonal elements) of the real rotation matrix given above is {{math|1 + 2 cos ''φ''}}. Since a trace is invariant under an orthogonal matrix similarity transformation,
:<math>
\mathrm{Tr}\left[\mathbf{A} \mathbf{R} \mathbf{A}^\mathsf{T}\right] =
\mathrm{Tr}\left[ \mathbf{R} \mathbf{A}^\mathsf{T}\mathbf{A}\right] = \mathrm{Tr}[\mathbf{R}]\quad\text{ with }\quad \mathbf{A}^\mathsf{T} = \mathbf{A}^{-1},
</math>
it follows that all matrices that are equivalent to {{math|'''R'''}} by such orthogonal matrix transformations have the same trace: the trace is a ''class function''. This matrix transformation is clearly an [[equivalence relation]], that is, all such equivalent matrices form an equivalence class. 

In fact, all proper rotation {{nowrap|3 × 3}} rotation matrices form a [[group (mathematics)|group]], usually denoted by SO(3) (the special orthogonal group in 3 dimensions) and all matrices with the same trace form an equivalence class in this group. All elements of such an equivalence class ''share their rotation angle'', but all rotations are around different axes. If {{math|'''n'''}} is an eigenvector of {{math|'''R'''}} with eigenvalue 1, then {{math|'''An'''}} is also an eigenvector of {{math|'''ARA'''}}<sup>T</sup>, also with eigenvalue 1. Unless {{math|'''A''' {{=}} '''I'''}}, {{math|'''n'''}} and {{math|'''An'''}} are different.