CHAPTER 3 - Random Matrices and Random Algebraic Polynomials

Abstract

This chapter discusses random matrices, focusing on some relations between random matrices and the random algebraic polynomials associated with them. Random matrices can be defined simply as matrices of which elements are random variables or random functions. The chapter presents some other definitions of random matrices. It also presents some examples of random matrices that arise in various applied fields, such as random matrices in mathematical statistics, quantum mechanics, and mathematical economics and random matrices associated with finite Markov chains. The chapter discusses random matrices and their associated random characteristic polynomials. The chapter further presents Newton's formula for random algebraic polynomials and explains the usefulness of Newton's formula. It also discusses companion matrices to random algebraic polynomials.