Complex symmetric nilpotent matrix ($N=\left(\begin{array}{cc}1& i\\ i& -1\end{array}\right),N^2=0$)

The displayed nonzero matrix is symmetric under transpose but not Hermitian. It is nilpotent, so all its eigenvalues are zero and it cannot be diagonalizable. Thus real-symmetric spectral conclusions require reality or a Hermitian hypothesis; complex transpose symmetry alone is insufficient.