I'm trying to find out a normal, real and $\boldsymbol n\times \boldsymbol n$ ($n\ge3$) matrix $A$ which is diagonalize over $C$ but isn't diagonalize over $R$.
I know that the following matrix (a.k.a the rotating matrix) satisfies the conditions above and isn't diagonalize over $R$:
But even within this example in my mind, I can't find such a matrix (satisfies the conditions above and isn't diagonalize over $R$) for $n\ge3$.
I'm almost sure there exists such a matrix. Can please someone give me an hint on how to establish such a matrix?