  We diagonalize numerically a Fibonacci matrix with fractal Hilbert space
structure of dimension $d_{f}=1.8316...$ We show that the density of states is
logarithmically normal while the corresponding level-statistics can be
described as critical since the nearest-neighbor distribution function
approaches the intermediate semi-Poisson curve. We find that the eigenvector
amplitudes of this matrix are also critical lying between extended and
localized.
