Let $v_1, v_2$ be generalized eigenvectors of ranks $l_1, l_2 > 0$, which respectively belong to eigenvalues $\lambda_1 \neq \lambda_2$ of the linear operator $A$:
$(A - \lambda_i I)^{l_i} v_i = 0$ and $(A - \lambda_i I)^{l_i - 1} v_i \neq 0$
The standard proof for usual eigenvectors (e.g., here) assumes that $A$ is symmetric.
Question: Must $v_1$ and $v_2$ be orthogonal? For an arbitrary linear operator $A$? If not, I'd be glad to a counter-example. What conditions on $A$ guarantee that $v_1$ and $v_2$ are orthogonal?
NB Base field may be taken as reals or complex, don't mind.
