Random matrix theory

Acta Numerica (2005), pp. 1–65 c Cambridge University Press, 2005DOI: 10.1017/S0962492904000236 Printed in the United KingdomRandom matrix theoryAlan EdelmanDepartment of Mathematics,Massachusetts Institute of Technology,Cambridge, MA 02139, USAE-mail: edelman@math.mit.eduN. Raj RaoDepartment of Electrical Engineering and Computer Science,Massachusetts Institute of Technology,Cambridge, MA 02139, USAE-mail: raj@mit.eduRandom matrix theory is now a big subject with applications in many discip-lines of science, engineering and finance. This article is a survey specificallyoriented towards the needs and interests of a numerical analyst. This sur-vey includes some original material not found anywhere else. We include theimportant mathematics which is a very modern development, as well as thecomputational software that is transforming the theory into useful practice.CONTENTS1 Introduction 22 Linear systems 23 Matrix calculus 34 Classical random matrix ensembles 115 Numerical algorithms stochastically 226 Classical orthogonal polynomials 257 Multivariate orthogonal po 308 Hypergeometric functions of matrix argument 329 Painlev´e equations 3310 Eigenvalues of a billion by billion matrix 4311 Stochastic operators 4612 Free probability and infinite random matrices 5113 A random matrix calculator 5314 Non-Hermitian and structured random matrices 5615 A segue 58References 592 A. Edelman and N. R. Rao1. IntroductionTexts on ‘numerical methods’ ...