A further consequence of equality in Eq. 3 is that except for a possible discrepancy associated with zero, the eigenvalues and eigenvectors of the connectivity matrix associated with a homomorphic image are the same as for the diagram itself. Suppose that A and B are the matrices and that z is an eigenvector of B.
Then
making Xz an eigenvector of A with the same eigenvalue. A similar
relation, valid in the opposite direction, holds for the row
eigenvectors. Precaution is necessary when 
vanishes, as it
must when A and B have different dimensions; the numbers of
eigenvectors associated with zero will differ.