Editing Orthogonal matrix (section)

===Canonical form===
More broadly, the effect of any orthogonal matrix separates into independent actions on orthogonal two-dimensional subspaces. That is, if {{mvar|Q}} is special orthogonal then one can always find an orthogonal matrix {{mvar|P}}, a (rotational) [[change of basis]], that brings {{mvar|Q}} into block diagonal form:

<math display="block">P^\mathrm{T}QP = \begin{bmatrix}
R_1 & & \\ & \ddots & \\ & & R_k
\end{bmatrix}\ (n\text{ even}),
\ P^\mathrm{T}QP = \begin{bmatrix}
R_1 & & & \\ & \ddots & & \\ & & R_k & \\ & & & 1
\end{bmatrix}\ (n\text{ odd}).</math>

where the matrices {{math|''R''<sub>1</sub>, ..., ''R''<sub>''k''</sub>}} are {{nowrap|2 × 2}} rotation matrices, and with the remaining entries zero. Exceptionally, a rotation block may be diagonal, {{math|±''I''}}. Thus, negating one column if necessary, and noting that a {{nowrap|2 × 2}} reflection diagonalizes to a +1 and −1, any orthogonal matrix can be brought to the form
<math display="block">P^\mathrm{T}QP = \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>

The matrices {{math|''R''<sub>1</sub>, ..., ''R''<sub>''k''</sub>}} give conjugate pairs of eigenvalues lying on the unit circle in the [[complex number|complex plane]]; so this decomposition confirms that all [[Eigenvalues and eigenvectors|eigenvalues]] have [[absolute value]] 1. If {{mvar|n}} is odd, there is at least one real eigenvalue, +1 or −1; for a {{nowrap|3 × 3}} rotation, the eigenvector associated with +1 is the rotation axis.