votes up vote 2 down vote accepted Clearly, if $\cos\phi=\pm1$, we must have $F=\pm id$. Thus we may consider only the case $\cos\phi\not=\pm1$. A real normal matrix $A$ can be orthogonally block-diagonalized. More specifically, there exists orthogonal matrix $Q$ such that $QAQ^T = (|\lambda_1|R(\theta_1)) \oplus\ldots\oplus(|\lambda_k|R(\theta_k))\oplus \Lambda$, where each $\lambda_i$ is some