# Universal eigenvalue statistics and vibration response prediction

## Abstract

It is well known that, under broad conditions, the local statistics of the eigenvalues of a random matrix tend towards a universal distribution if the matrix is sufficiently random, regardless of the statistics of the matrix entries. This distribution is the same as that associated with a class of random matrix known as the Gaussian Orthogonal Ensemble (GOE). The underlying reason for this behaviour is explored here, and it is concluded that the source of the universal statistics lies in the Vandermonde determinant, which appears in the Jacobian of the transformation between the entries of a matrix and its eigenvalues and eigenvectors. Attention is then turned to the application of this result to natural frequency statistics, and to the prediction of the response statistics of a random built-up system. Recent work is reviewed in which it is shown that, away from low frequencies, the mean and variance of the system response can be predicted in an efficient way without knowledge of the statistics of the underlying system uncertainties.

## Keywords

Acoustical Society Random Matrix Modal Density Random Matrix Theory Joint Probability Density Function## Preview

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