Canonical Equation K ₄₉ for Normalized Spectral Functions of a Product of Random Unitary Matrices

Abstract

In this chapter, we analyze the spectral properties of the product of unitary independent matrices in the case where the number of matrices increases and every matrix, in some probability sense, converges to the identity matrix of increasing order. Such matrices resemble the unitary matrizant, which converges to a matrix of infinite dimension. Such matrices are used in many applied sciences, especially in the theory of gyroscopes, control theory and physics. In this section, we follow the method developed by V. A. Marchenko and L. A. Pastur [MaP]. This method is based on the differences of two normalized traces of resolvents of random matrices and on the derivation of a certain partial differential equation