Canonical Equation K ₁

Abstract

Starting from random symmetric matrices, we deduce the first canonical equation by introducing a procedure of averaging over random entries of random matrices. Numerous works are devoted to the analysis of the spectral functions of random matrices (see the reviews and books devoted to the spectral theory of random matrices in the bibliography of the present book); however, no one has managed to solve the problem of deducing an equation for the Stieltjes transform of the spectral functions of large-size random symmetric matrices whose entries on the principal diagonal and above it are independent and their variances are bounded by a constant. In this chapter, we propose a method for the solution of this problem by using the formulas suggested by the author in [Gir12]. Since this equation holds for general random matrices and is often used in applied research, we call it, a canonical spectral equation. In this chapter, we assume that the distribution of the entries of random matrices is not normal and use the method of martingale differences.