This is a question in elementary linear algebra, though I hope it's not so trivial to be closed.

Real symmetric matrices, complex hermitian matrices, unitary matrices, and complex matrices with distinct eigenvalues are diagonalizable, i.e. conjugate to a diagonal matrix.

I'd just like to see an example of a complex symmetric $n\times n$ matrix that is not diagonalizable.

similarto a complex symmetric one, but not necessarily unitarily so. (See the comments on page 1 of arxiv.org/abs/0907.2728v2 ) In particular, any non-trivial nilpotent would do. – Yemon Choi May 5 '10 at 21:36