Eigenvalues and eigenvectors

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.

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Harold V. McIntosh
E-mail:mcintosh@servidor.unam.mx