I was wondering given the Hermitian matrix: $$H =\begin{bmatrix} A & B\\ C & D \end{bmatrix}$$ Are the eigenvectors of $A$ and $D$ at all related to the eigenvectors of $H$? In particular, I want to approximately find the eigenvector of H with the smallest eigenvalue through a variational algorithm (H is the hamiltonian for a quantum system and thus has a very large dimension), and I was wondering if there's any mathematical reasoning to be made as to why using the eigenvectors of block matrices would be a good heuristic.
I'm not looking for a proof or a definite formula (one likely doesn't exist) But rather any sort of relationship, or an intuitive reason as to whether or not the eigenvectors of A are any closer to the eigenvectors of H than a randomly chosen vector. (Here, by eigenvectors of A I mean the padded with $0$ version to fit the dimension e.g ($v1,...,v_n, 0_{n + 1},..0_{2n}$))