  We numerically analyze the statistical properties of complex system with
conditions subjecting the matrix elements to a set of specific constraints
besides symmetry, resulting in various structures in their matrix
representation. Our results reveal an important trend: while the spectral
statistics is strongly sensitive to the number of independent matrix elements,
the eigenfunction statistics seems to be affected only by their relative
strengths. This is contrary to previously held belief of one to one relation
between the statistics of the eigenfunctions and eigenvalues (e.g. associating
Poisson statistics to the localized eigenfunctions and Wigner-Dyson statistics
to delocalized ones).
