Given positive semidefinite matrices $A,B \succeq 0$, $A, B \in \mathbb{R}^{n \times n}$, if we have $$\langle A, B \rangle = 0,$$ where $\langle \cdot, \cdot \rangle$ denotes the Frobenius inner product, then

What are tight necessary conditions of ranks of $A, B$? For example, $\mbox{rank} (A) + \mbox{rank}(B) \leq n$? How to prove that?

What can be say about the orthogonality of the eigenvectors of $A$ and $B$? For example, are they pairwise orthogonal?