Assume $A$ is a complex $n \times n$ matrix such that $$\forall \lambda \in \mathbb{C} \, : \, \ker(\lambda I-A)$$ is the orthogonal complement of $\operatorname{range}(\lambda I-A)$.
Show that there exists an orthonormal basis of $\mathbb{C}^n$ consisting of eigenvectors of $A$.
I tried to show $A$ is normal using the fact that $\ker(\lambda I-A)=\ker(\bar\lambda I-A^*)$ but I was unsuccessful.
One approach is as follows. With a Schur-decomposition, we can assume without loss of generality that $A$ is upper triangular. Now, show that if $A$ is upper triangular and satisfies $\ker(\lambda I - A) = \ker(\bar \lambda - A^*)$, then it must be diagonal (and hence normal).