If two antisymmetric and symmetric matrices are similar, why are they zero?
My work : Let $A$ be the antisymmetric matrix and $S$ the symmetric one. $A$ is diagonalizable in $M_n(\mathbb C)$, we write $\Delta_A$ its diagonal matrix. $S$ is diagonalizable in $M_n(\mathbb R)$, we write $\Delta_S$ its diagonal matrix.
$\Delta_A$ and $\Delta_S$ are similar, so they must have the same eighten values : it is impossible because some are in $\mathbb R$, the others in $\mathbb iR$. So both matrixes must be zero