Editing Orthogonal group (section)

=== Canonical form ===
For any element of {{math|O(''n'')}} there is an orthogonal basis, where its matrix has the form
: <math>\begin{bmatrix}
  \begin{matrix}
    R_1 &        & \\
        & \ddots & \\
        &        & R_k
  \end{matrix} & 0 \\
  0 & \begin{matrix}
    \pm 1 &        & \\
          & \ddots & \\
          &        & \pm 1
  \end{matrix}\\
\end{bmatrix},</math>
where there may be any number, including zero, of ±1's; and where the matrices {{math|''R''<sub>1</sub>, ..., ''R''<sub>''k''</sub>}} are 2-by-2 rotation matrices, that is matrices of the form 
: <math>\begin{bmatrix}a&-b\\b&a\end{bmatrix},</math>
with {{math|1=''a''{{sup|2}} + ''b''{{sup|2}} = 1}}.

This results from the [[spectral theorem]] by regrouping [[eigenvalues]] that are [[complex conjugate]], and taking into account that the absolute values of the eigenvalues of an orthogonal matrix are all equal to {{math|1}}.

The element belongs to {{math|SO(''n'')}} if and only if there are an even number of {{math|−1}} on the diagonal. A pair of eigenvalues {{math|−1}} can be identified with a rotation by {{math|π}} and a pair of eigenvalues {{math|+1}} can be identified with a rotation by {{math|0}}.

The special case of {{math|1=''n'' = 3}} is known as [[Euler's rotation theorem]], which asserts that every (non-identity) element of {{math|SO(3)}} is a [[rotation]] about a unique axis–angle pair.