Is a complex symmetric square matrix with zero diagonal diagonalizable?

Question

Let $A$ be a complex symmetric square matrix, and set the diagonal to 0. Thus $a_{ii} = 0$ for all $1\leq i \leq n$. I am wondering if $A$ is always diagonalizable. I read that almost all complex symmetric square matrices are diagonalizable, and thought that perhaps the zero diagonal would simplify things sufficiently to allow a proof for arbitrary $A$.

Answer 1

Take$$A=\begin{pmatrix}0 & 0 & i \\ 0 & 0 & 1 \\ i & 1 & 0\end{pmatrix}.$$Its characteristic polynomial is $-x^3$ and therefore, if it was diagonalizable, it would be the null matrix. But it isn't. Therefore, $A$ is a complex symmetric matrix with only $0$'s in the main diagonal which is not diagonalizable.

Answer 2

page 209 in Horn and Johnson, Matrix Analysis, say (well, it is part of something else) $$ \left( \begin{array}{rrrr} 0 & 1 & -i & 0 \\ 1 & 0 & 0 & i \\ -i & 0 & 0 & 1 \\ 0 & i & 1 & 0 \end{array} \right) $$ is not similar to a diagonal matrix. Worth checking by hand. The characteristic polynomial is $x^4,$ the minimal polynomial is $x^2.$ So Jose's argument applies.

Let me add that a symmetric complex matrix $M$ is congruent to a diagonal matrix: there is a matrix $P$ with $\det P \neq 0$ so that $P^T MP = D$ is diagonal.