Methods are developed for calculating probabilistic characteristics of the eigenvalues of stochastic symmetric matrices. The methods are based on the relationship between the elements of a matrix and its eigenvalues, perturbation method, bounds on eigenvalues, and zero-crossings of the characteristic polynomial. It is shown that the polynomial characteristic of a stochastic matrix can be viewed as a random process whose crossings of level zero define the eigenvalues of the matrix. The proposed methods of analysis are demonstrated by examples from dynamics and elasticity.

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