Are there diagonalisable endomorphisms which are not unitarily diagonalisable? I know that normal endomorphisms are unitarily diagonalisable. Now I'm wondering, are there any diagonalisable endomorphisms which are not unitarily diagonalisable? 
If so, could you provide an example? 

Another way to look at it, though no really different in essence, is to consider the operator norm on ${\rm M}_{n}(\mathbb{C})$ induced by the Euclidean norm on $\mathbb{C}^{n}$ (thought of as column vectors). Hence $\|A \| = {\rm max}_{ v : \|v \| = 1} \|Av \|.$ Since the unitary transformations are precisely the isometries of $\mathbb{C}^{n},$ we see that conjugation by a unitary matrix does not change the norm of a matrix. If $A$ can be diagonalized by a unitary matrix, then it is clear from this discussion that $\| A \| = {\rm max}_{\lambda} |\lambda |,$ as $\lambda$ runs over the eigenvalues of $A$. Hence as soon as we find a diagonalizable matrix $B$ with $\| B \| \neq {\rm max}_{\lambda} |\lambda|,$ we know that $B$ is not diagonalizable by a unitary matrix. For example, the matrix $$B = \left( \begin{array}{clcr} 3 & 5\\0 & 2 \end{array} \right)$$ has largest eigenvalue $3,$ but $\|B \| > 5$  because
$B \left( \begin{array}{cc} 0 \\1 \end{array} \right) = \left( \begin{array}{cc} 5 \\2 \end{array} \right).$ Also, $B$ is diagonalizable, but we now know that can't be achieved via a unitary matrix.
