Does Eigen support calculation of eigenvalues of a sparse complex self-adjoint matrix? When I tried using SelfAdjointEigenSolver with such a matrix I get many errors!
Another question, what is the default storage scheme for the sparse matrix: Column major or row-major and one-based or zero-based indexing ?

Unlike the name suggest, Eigen is not a specialized library for eigenvalue problems! Currently, there is no support for sparse eigenvalue decomposition, but a search in this forum will give you various pointers to alternatives. Beside, by default SparseMatrix is column major (but you can create a row major matrix too) with zero-based indexing.