Let $A$ be a real $n\times n$ matrix. Can $A$ be normal, diagonalizable over $\mathbb{R}$ but not unitary diagonalizable over $\mathbb{R}$ ? By "unitary diagonalizable over $\mathbb{R}$" I mean that there exist a real unitary matrix $P$ such that $P^tAP$ is diagonal.
My attempt: I need to find a non-symmetric real matrix which is normal ?
No. If $A$ is normal, it is unitarily diagonalisable over $\mathbb C$. Let $A=UDU^\ast$ be such a diagonalisation. Since $A$ is supposed to be diagonalisable over $\mathbb R$, $D$ is a real diagonal matrix. Therefore $A$ is Hermitian. Yet, $A$ is real. Hence it is real symmetric and unitarily/orthogonally diagonalisable over $\mathbb R$.