Closed-form expressions for eigenvalue and eigenvectors of stochastic symmetric matrices using the probability transformation method

Abstract

This work shows the use of the Probability Transformation Method (PTM) for deriving a closed-form probability density function (PDF) of the eigenpair of stochastic real-valued symmetric matrices. In particular, the PTM allows the direct evaluation of the eigenpair PDF starting from the joint PDF (JPDF) of the system’s uncertainties. The impact of the linear stochastic systems’ randomness in the natural frequencies and mode shape is investigated through some numerical applications. Even if the structural samples investigated are intentionally simple, that aspect is only linked to the authors’ use of the Mathematica software that, in some ways, limits the resolution for high dimensional problems. From a theoretical perspective, though, this is not a restriction, and the problem’s dimension has no impact on the method’s accuracy. The obtained analytical results compared with Monte Carlo simulations have confirmed the goodness of the proposed stochastic procedure.