Canonical Equation K 51 for Normalized Spectral Functions of a Product of Random Independent Matrices

Abstract

Beginning with this chapter, we start to analyze the spectral properties of a product of independent random matrices in the case where the number of matrices increases and every matrix, in a certain probability sense, converges to the identity matrix. Our theory is completely different as compared with the corresponding spectral theory for Hermitian or unitary random matrices. We develop it on the basis of the so-called V-transform. Such matrices resemble the matrizant, which converges to a matrix of infinite dimension. They are used in many applied sciences, especially in mechanics, control theory and physics. For example, the system of differential linear equations $$ \frac{{d{{\vec x}_n}(t)}}{{dt}}\; = \;{\Xi _{n\;{\rm{x }}n}}\;{\rm{(}}t{\rm{) }}{\vec x_n}\;{\rm{(}}t{\rm{)}},\;{\rm{0 }} \le {\rm{ }}t \le {\rm{ }}T,{\rm{ }}{\vec x_n}\;{\rm{(}}0{\rm{) = }}\vec c $$ is very often considered; here, Ξ n x n (t) is a random matrix process such that the following matrices \( {\Xi _{n\;{\rm{x }}n}}\;(t)\;{\rm{ = }}H_{n\;{\rm{x }}n}^{(i)},\;{t_{i - 1}}\; \le \;t\; < {t_i},\;i\;{\rm{ = 1, }} \ldots {\rm{, }}m{\rm{ }} \) are independent, where 0 = t 0 < t 1 < ··· < t m = T. Then the solution of this system is equal to $$ {\vec x_n}\;(T)\;{\rm{ = }}\mathop \Pi \limits_{i = 1}^m \;{\rm{exp }}\left\{ {\left. {{\rm{(}}{t_i}{\rm{ - }}{t_{i - 1}}{\rm{) }}H_{n\;{\rm{x }}n}^{(i)}} \right\}} \right.\vec c, $$ and we obtain the product of independent non-Hermitian matrices. In this chapter we consider some simple examples of such products.