Calculating eigen values of very large sparse matrices in python
I agree with @pv. If your matrix S was symmetric, you could see it as a laplacian matrix of the matrix I - S. The number of connected components of I - S is the number of zero-eigenvalues of this matrix (i.e, the dimension of the space associated to eigenvalue 1 of S). You could check the number of connected components of the graph whose similarity matrix is I - S*S' for a start, e.g. with scipy.sparse.csgraph.connected_components.