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.



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> I'd just like to see an example of a complex symmetric $n\times n$ matrix that is not diagonalizable.
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Non-diagonalizable complex symmetric matrix
$$\begin{pmatrix} 1 & i \\ i & -1 \end{pmatrix}.$$


How did I find this? Non-diagonalizable means that there is some Jordan block of size greater than $1$. I decided to hunt for something with Jordan form $\left( \begin{smallmatrix} 0 & 1 \\ 0 & 0 \end{smallmatrix} \right)$. So I want trace and determinant to be zero, but the matrix not to be zero. The diagonal entries made sure the trace vanished, and then the off diagonal entries were forced.


