Consider a finite dimensional block diagonal matrix $A$ over $\mathbb{C}$ given by $$ A = \bigoplus_i A_i, $$ where $A_i$ are mostly the 0 matrix (of some finite dimension), i.e. the matrix is very sparse.
Is there a nice property of block diagonal matrices that lets me independently diagonalise the $A_i$'s and then somehow glue the result to diagonalise the full $A$?
(I am numerically solving for the eigenvectors of a large sparse matrix and I am running out of memory and sparse coding representations don't help)