TITLE: Show any orthogonal matrix is similar to an almost diagonal matrix, with either $\pm 1$ or a 2D rotation on the diagonal
QUESTION [8 upvotes]: Let $A \in O(n).$  Show that $A$ is similar to a matrix which consists of $2 \times 2$ blocks down the diagonal of the form
$$ \begin{pmatrix} \cos{\theta} & \sin{\theta}\\-\sin{\theta} & \cos{\theta} \end{pmatrix},$$
along with some diagonal elements which are $+1$ and $-1.$ 
For example, maybe $A \in O(5)$ is similar to,
$$\begin{pmatrix} 
1 & 0 & 0 & 0 & 0\\
0 & \cos{\theta} & \sin{\theta} & 0 & 0\\
0 & -\sin{\theta} & \cos{\theta} & 0 & 0\\
0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & -1
\end{pmatrix}.$$
This makes intuitive sense to me, though I am not sure how I'd go about the proof.  I've been trying to use a subspace invariant method, like $W$ is a subspace of $\mathbb{R^n}$ that is $A$-invariant, then its orthogonal complement is too, but it hasn't gotten me much closer.  Any hints/solutions?

REPLY [2 votes]: Some hints which may be useful
All eigenvalues will come in complex conjugate pairs ( why? ). Furthermore for orthogonality $|\lambda_k| = 1$ for all $k$, can we show a $2\times 2$ $$\left[\begin{array}{cc}\lambda_l&0\\0&\overline \lambda_l\end{array}\right] \text{similar to } \left[\begin{array}{cc}\cos(\alpha_l)&\sin(\alpha_l)\\-\sin(\alpha_l)&\cos(\alpha_l)\end{array}\right]$$
If that is so, we can first do diagonalization over $\mathbb{C}$ and then map all such pairs $\lambda , \overline \lambda$ onto $2\times 2$ rotation matrices by a set of similarity transforms.