  We consider the Gaussian ensembles of random matrices and describe the normal
modes of the eigenvalue spectrum, i.e., the correlated fluctuations of
eigenvalues about their most probable values. The associated normal mode
spectrum is linear, and for large matrices, the normal modes are found to be
Chebyshev polynomials of the second kind. We contrast this with the behaviour
of a sequence of uncorrelated levels, which has a quadratic normal mode
spectrum. The difference in the rigidity of random matrix spectra and sequences
of uncorrelated levels can be attributed to this difference in the normal mode
spectra. We illustrate this by calculating the number variance in the two
cases.
