This paper uses a technique of
abstract spectral theory to reduce the study of certain eigenvalues, which are not
necessarily isolated, to the case of isolated eigenvalues. By this method the
Weinstein-Aronszajn formula for the change in multiplicity of an isolated eigenvalue
of a self adjoint operator under a finite dimensional perturbation is extended. The
hypotheses of this generalization are studied in the abstract and also by
demonstrative example.