Random matrix theory, random point process theory, and natural frequency statistics

Abstract

There has been much recent interest in the application of random matrix theory to the dynamics of uncertain engineering structures. In particular, it has been found experimentally and numerically that the results arising from the Gaussian orthogonal ensemble (GOE) are applicable to natural frequency spacing statistics. The occurrence of natural frequencies can also be viewed as a random point process, and although there is much in common between random point process theory and random matrix theory, the two subjects appear to have been developed independently. It is shown here that the two approaches employ very similar mathematical functions, albeit under a different name: For example, the distribution function (point process theory) corresponds to the n-point correlation function (random matrix theory), and the cumulant function corresponds to the n-level cluster function. By recognizing this similarity, it is possible to apply a number of established results in random point process theory to the statistics of the eigenvalues of a random matrix. This leads to new insights into the statistics of natural frequency spacings, and helps to explain why the Wigner surmise (which states that the spacings have a Rayleigh distribution) is applicable to a much wider class of matrix than the GOE.