  A generalization of Brillouin-Wigner perturbation theory is applied
numerically to the Wigner Band Random Matrix model. The perturbation theory
tells that a perturbed energy eigenstate can be divided into a perturbative
part and a non-perturbative part with the perturbative part expressed as a
perturbation expansion. Numerically it is found that such a division is
important in understanding many properties of both eigenstates and the
so-called local spectral density of states (LDOS). For the average shape of
eigenstates, its central part is found to be composed of its non-perturbative
part and a region of its perturbative part, which is close to the
non-perturbative part. A relationship between the average shape of eigenstates
and that of LDOS can be explained. Numerical results also show that the
transition for the average shape of LDOS from the Breit-Wigner form to the
semicircle form is related to a qualitative change in some properties of the
perturbation expansion of the perturbative parts of eigenstates. The transition
for the half-width of the LDOS from quadratic dependence to linear dependence
on the perturbation strength is accompanied by a transition of a similar form
for the average size of the non-perturbative parts of eigenstates. For both
transitions, the same critical perturbation strength $\lambda_b$ has been found
to play important roles. When perturbation strength is larger than $\lambda_b$,
the average shape of LDOS obeys an approximate scaling law.
