TITLE: Diagonalizable vs Normal
QUESTION [5 upvotes]: If I have real valued matrix $A$, are these two notions of being Diagonalizable and being Normal equivalent?

REPLY [7 votes]: A real matrix is diagonalizable (over $\mathbb{C}$) by a unitary matrix if and only if it is normal.
But there are non-normal matrices that can be diagonalized by non-unitary matrices. For instance
$$\left[\begin{array}{cc}0 & 1\\2 & 1\end{array}\right] = PDP^{-1}$$
where 
$$P = \left[\begin{array}{cc} -1 & 1\\ 1& 2\end{array}\right]$$
 and 
$$D=\left[\begin{array}{cc} -1 &0\\0 & 2\end{array}\right].$$