Nonparametric stochastic modeling of linear systems with prescribed variance of several natural frequencies

A complete probabilistic model of random positive definite matrices is developed that incorporates constraints on the standard deviations of a set of its eigenvalues. The model is, in particular, applicable to the representation of the mass and stiffness matrices of random dynamic systems of which certain natural frequencies are observed. The model development is based on the maximization of the entropy under a set of constraints representing the prescribed eigenvalue standard deviations, the mean matrix being given, and the existence of the mean Frobenius norm of the inverse of the random matrix. The efficient simulation of samples of random matrices according to the proposed model is discussed in detail. Finally, examples of application validate the above concepts and demonstrate the usefulness of the proposed model.