$A \in \mathbb{C}^{n \times n }$ is skew hermitian if $A^*=-A $. Show that eigenvalue of skew hermitian matrices are purely imaginary.
Try:
Let $v \in \mathbb{C}^n$ be eigenvector of $A$ with associated eigenvalue $\lambda$. We know $$(Av)^* v = v^* A^* v = v^* (-A) v = - v^* (Av)=-v^* \lambda A= - \lambda v^* v$$
On the other hand,
$$ (Av)^* v = (\lambda v)^* v = \overline{\lambda} v^* v $$
And so $- \lambda = \overline{\lambda}$ so $\lambda$ is purely imaginary. IS this correct appraoch ?
