Variations in strength, geometric and deterioration characteristics of both materials and machine components are quite common in real-life mechanical systems due to manufacturing defects and measurement errors. Such inherent fluctuations which are unavoidable even with the best quality control measures, are essentially random in nature. Effects of these random fluctuations on the performance levels, dynamic response and service life of mechanical systems need to be evaluated based on a stochastic approach, in order to assist design and diagnostics of industrial machinery. Non self-adjoint eigenproblems that correspond to the dynamic response of complex mechanical systems such as high speed rotors, fluid-flowing pipes and actively controlled structures are considered in the present work. The coefficients of the matrices are stochastic processes and are resulting from uncertain parameters of the mechanical system being described by the eigenproblem. A perturbational solution is sought and obtained in a form that does not involve repeated solutions of a recursive set of equations. Sample functions are generated based on the perturbational expansion and response moments are obtained by treating uncertain fluctuations to be stochastic perturbations. Complete covariance structures of both eigenvalues and eigenvectors are obtained through computationally efficient expressions. Applications of the developed procedure for real-life mechanical systems, that have uncertain material properties, are demonstrated.