Given a hermitian matrix $H$ as follows: \begin{equation} H = \begin{bmatrix} H^1 & V^{12} \\ V^{21} & H^2 \end{bmatrix}. \end{equation} Here, $H^1,H^2\in\mathbb{C}^{N\times N}$ are diagonal (or block-diagonal) matrices, $V^{12}={V^{21}}^\dagger \in \mathbb{C}^{N\times N}$. When N is very large, is there exists some methods allowing me to get the approximated eigenvalues and eigenvectors of $H$?
Intuitively, it seems the diagonal structure of $H^1,H^2$ is helpful, maybe by constructing an isomorphism between $H$ and the tensor product $H^1 \otimes H^2$ and then docomposite $H^1$ or $H^2$ as a direct sum: \begin{equation} H^1 \otimes H^2 = \oplus_{i}(H^1_i\otimes H^2)=\oplus_{j}(H^1\otimes H^2_j). \end{equation} But I am not sure whether it is truely helpful. And I don't know how to constructe such isomorphism with a general $V^{12}$.