TITLE: Show that orthogonal matrices are diagonalizable
QUESTION [5 upvotes]: I want to prove that all orthogonal matrices are diagonalizable over $C$. I know that a matrix is orthogonal if $Q^TQ = QQ^T = I$ and $Q^T = Q^{-1}$, and that a matrix $A$ is diagonalizable if $A = PDP^{-1}$ where $D$ is a diagonal matrix. How can I start this proof?

REPLY [4 votes]: Note that if $Q$ is orthogonal then $Q$ is normal, because
\begin{equation*}
Q Q^T = Q^T Q = I.
\end{equation*}
So the spectral theorem implies that $Q$ is diagonalizable.