Preface to the Second Edition...vii Preface to the First Edition... ix
|
|
- Kathleen Francis
- 6 years ago
- Views:
Transcription
1 Contents Preface to the Second Edition...vii Preface to the First Edition ix 1 Introduction Large Dimensional Data Analysis Random Matrix Theory Spectral Analysis of Large Dimensional Random Matrices Limits of Extreme Eigenvalues Convergence Rate of the ESD Circular Law CLT of Linear Spectral Statistics Limiting Distributions of Extreme Eigenvalues and Spacings Methodologies Moment Method Stieltjes Transform Orthogonal Polynomial Decomposition Free Probability Wigner Matrices and Semicircular Law Semicircular Law by the Moment Method Moments of the Semicircular Law Some Lemmas in Combinatorics Semicircular Law for the iid Case Generalizations to the Non-iid Case Proof of Theorem Semicircular Law by the Stieltjes Transform Stieltjes Transform of the Semicircular Law Proof of Theorem xi
2 xii Contents 3 Sample Covariance Matrices and the Marčenko-Pastur Law M-P Law for the iid Case Moments of the M-P Law Some Lemmas on Graph Theory and Combinatorics M-P Law for the iid Case Generalization to the Non-iid Case Proof of Theorem 3.10 by the Stieltjes Transform Stieltjes Transform of the M-P Law Proof of Theorem Product of Two Random Matrices Main Results Some Graph Theory and Combinatorial Results Proof of Theorem Truncation of the ESD of T n Truncation, Centralization, and Rescaling of the X-variables Completing the Proof LSD of the F-Matrix Generating Function for the LSD of S n T n Completing the Proof of Theorem Proof of Theorem Truncation and Centralization Proof by the Stieltjes Transform Limits of Extreme Eigenvalues Limit of Extreme Eigenvalues of the Wigner Matrix Sufficiency of Conditions of Theorem Necessity of Conditions of Theorem Limits of Extreme Eigenvalues of the Sample Covariance Matrix Proof of Theorem Proof of Theorem Necessity of the Conditions Miscellanies Spectral Radius of a Nonsymmetric Matrix TW Law for the Wigner Matrix TW Law for a Sample Covariance Matrix Spectrum Separation What Is Spectrum Separation? Mathematical Tools Proof of (1) Truncation and Some Simple Facts A Preliminary Convergence Rate
3 Contents xiii Convergence of s n Es n Convergence of the Expected Value Completing the Proof Proof of (2) Proof of (3) Convergence of a Random Quadratic Form spread of eigenvaluesspread of Eigenvalues Dependence on y Completing the Proof of (3) Semicircular Law for Hadamard Products Sparse Matrix and Hadamard Product Truncation and Normalization Truncation and Centralization Proof of Theorem 7.1 by the Moment Approach Convergence Rates of ESD Convergence Rates of the Expected ESD of Wigner Matrices Lemmas on Truncation, Centralization, and Rescaling Proof of Theorem Some Lemmas on Preliminary Calculation Further Extensions Convergence Rates of the Expected ESD of Sample Covariance Matrices Assumptions and Results Truncation and Centralization Proof of Theorem Some Elementary Calculus Increment of M-P Density Integral of Tail Probability Bounds of Stieltjes Transforms of the M-P Law Bounds for b n Integrals of Squared Absolute Values of Stieltjes Transforms Higher Central Moments of Stieltjes Transforms Integral of δ Rates of Convergence in Probability and Almost Surely CLT for Linear Spectral Statistics Motivation and Strategy CLT of LSS for the Wigner Matrix Strategy of the Proof Truncation and Renormalization Mean Function of M n Proof of the Nonrandom Part of (9.2.13) for j = l, r.. 238
4 xiv Contents 9.3 Convergence of the Process M n EM n Finite-Dimensional Convergence of M n EM n Limit of S Completion of the Proof of (9.2.13) for j = l, r Tightness of the Process M n (z) EM n (z) Computation of the Mean and Covariance Function of G(f) Mean Function Covariance Function Application to Linear Spectral Statistics and Related Results Tchebychev Polynomials Technical Lemmas CLT of the LSS for Sample Covariance Matrices Truncation Convergence of Stieltjes Transforms Convergence of Finite-Dimensional Distributions Tightness of Mn 1 (z) Convergence of Mn(z) Some Derivations and Calculations Verification of (9.8.8) Verification of (9.8.9) Derivation of Quantities in Example (1.1) Verification of Quantities in Jonsson s Results Verification of (9.7.8) and (9.7.9) CLT for the F-Matrix CLT for LSS of the F-Matrix Proof of Theorem Lemmas Proof of Theorem CLT for the LSS of a Large Dimensional Beta-Matrix Some Examples Eigenvectors of Sample Covariance Matrices Formulation and Conjectures Haar Measure and Haar Matrices Universality A Necessary Condition for Property Moments of X p (F Sp ) Proof of (10.3.1) (10.3.2) Proof of (b) Proof of (10.3.2) (10.3.1) Proof of (c) An Example of Weak Convergence Converting to D[0, ) A New Condition for Weak Convergence
5 Contents xv Completing the Proof Extension of (10.2.6) to B n = T 1/2 S p T 1/ First-Order Limit CLT of Linear Functionals of B p Proof of Theorem Proof of Theorem An Intermediate Lemma Convergence of the Finite-Dimensional Distributions Tightness of Mn 1(z) and Convergence of M2 n (z) Proof of Theorem Circular Law The Problem and Difficulty Failure of Techniques Dealing with Hermitian Matrices Revisiting Stieltjes Transformation A Theorem Establishing a Partial Answer to the Circular Law Lemmas on Integral Range Reduction Characterization of the Circular Law A Rough Rate on the Convergence of ν n (x, z) Truncation and Centralization A Convergence Rate of the Stieltjes Transform of ν n (, z) Proofs of (11.2.3) and (11.2.4) Proof of Theorem Comments and Extensions Relaxation of Conditions Assumed in Theorem Some Elementary Mathematics New Developments Some Applications of RMT Wireless Communications Channel Models random matrix channelrandom Matrix Channels Linearly Precoded Systems Channel Capacity for MIMO Antenna Systems Limiting Capacity of Random MIMO Channels A General DS-CDMA Model Application to Finance A Review of Portfolio and Risk Management Enhancement to a Plug-in Portfolio A Some Results in Linear Algebra A.1 Inverse Matrices and Resolvent A.1.1 Inverse Matrix Formula A.1.2 Holing a Matrix
6 xvi Contents A.1.3 Trace of an Inverse Matrix A.1.4 Difference of Traces of a Matrix A and Its Major Submatrices A.1.5 Inverse Matrix of Complex Matrices A.2 Inequalities Involving Spectral Distributions A.2.1 Singular-Value Inequalities A.3 Hadamard Product and Odot Product A.4 Extensions of Singular-Value Inequalities A.4.1 Definitions and Properties A.4.2 Graph-Associated Multiple Matrices A.4.3 Fundamental Theorem on Graph-Associated MMs A.5 Perturbation Inequalities A.6 Rank Inequalities A.7 A Norm Inequality B Miscellanies B.1 Moment Convergence Theorem B.2 Stieltjes Transform B.2.1 Preliminary Properties B.2.2 Inequalities of Distance between Distributions in Terms of Their Stieltjes Transforms B.2.3 Lemmas Concerning Levy Distance B.3 Some Lemmas about Integrals of Stieltjes Transforms B.4 A Lemma on the Strong Law of Large Numbers B.5 A Lemma on Quadratic Forms Relevant Literature Index
7
Random Matrix Theory Lecture 1 Introduction, Ensembles and Basic Laws. Symeon Chatzinotas February 11, 2013 Luxembourg
Random Matrix Theory Lecture 1 Introduction, Ensembles and Basic Laws Symeon Chatzinotas February 11, 2013 Luxembourg Outline 1. Random Matrix Theory 1. Definition 2. Applications 3. Asymptotics 2. Ensembles
More informationClasses of Linear Operators Vol. I
Classes of Linear Operators Vol. I Israel Gohberg Seymour Goldberg Marinus A. Kaashoek Birkhäuser Verlag Basel Boston Berlin TABLE OF CONTENTS VOLUME I Preface Table of Contents of Volume I Table of Contents
More informationThe circular law. Lewis Memorial Lecture / DIMACS minicourse March 19, Terence Tao (UCLA)
The circular law Lewis Memorial Lecture / DIMACS minicourse March 19, 2008 Terence Tao (UCLA) 1 Eigenvalue distributions Let M = (a ij ) 1 i n;1 j n be a square matrix. Then one has n (generalised) eigenvalues
More informationContents. Preface for the Instructor. Preface for the Student. xvii. Acknowledgments. 1 Vector Spaces 1 1.A R n and C n 2
Contents Preface for the Instructor xi Preface for the Student xv Acknowledgments xvii 1 Vector Spaces 1 1.A R n and C n 2 Complex Numbers 2 Lists 5 F n 6 Digression on Fields 10 Exercises 1.A 11 1.B Definition
More informationLarge sample covariance matrices and the T 2 statistic
Large sample covariance matrices and the T 2 statistic EURANDOM, the Netherlands Joint work with W. Zhou Outline 1 2 Basic setting Let {X ij }, i, j =, be i.i.d. r.v. Write n s j = (X 1j,, X pj ) T and
More informationHands-on Matrix Algebra Using R
Preface vii 1. R Preliminaries 1 1.1 Matrix Defined, Deeper Understanding Using Software.. 1 1.2 Introduction, Why R?.................... 2 1.3 Obtaining R.......................... 4 1.4 Reference Manuals
More informationEstimation of the Global Minimum Variance Portfolio in High Dimensions
Estimation of the Global Minimum Variance Portfolio in High Dimensions Taras Bodnar, Nestor Parolya and Wolfgang Schmid 07.FEBRUARY 2014 1 / 25 Outline Introduction Random Matrix Theory: Preliminary Results
More informationWiley. Methods and Applications of Linear Models. Regression and the Analysis. of Variance. Third Edition. Ishpeming, Michigan RONALD R.
Methods and Applications of Linear Models Regression and the Analysis of Variance Third Edition RONALD R. HOCKING PenHock Statistical Consultants Ishpeming, Michigan Wiley Contents Preface to the Third
More informationMathematics for Economics and Finance
Mathematics for Economics and Finance Michael Harrison and Patrick Waldron B 375482 Routledge Taylor & Francis Croup LONDON AND NEW YORK Contents List of figures ix List of tables xi Foreword xiii Preface
More informationNo books, no notes, no calculators. You must show work, unless the question is a true/false, yes/no, or fill-in-the-blank question.
Math 304 Final Exam (May 8) Spring 206 No books, no notes, no calculators. You must show work, unless the question is a true/false, yes/no, or fill-in-the-blank question. Name: Section: Question Points
More informationMatrix Mathematics. Theory, Facts, and Formulas with Application to Linear Systems Theory. Dennis S. Bernstein
Matrix Mathematics Theory, Facts, and Formulas with Application to Linear Systems Theory Dennis S. Bernstein PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD Contents Special Symbols xv Conventions, Notation,
More informationHomework 1. Yuan Yao. September 18, 2011
Homework 1 Yuan Yao September 18, 2011 1. Singular Value Decomposition: The goal of this exercise is to refresh your memory about the singular value decomposition and matrix norms. A good reference to
More informationWigner s semicircle law
CHAPTER 2 Wigner s semicircle law 1. Wigner matrices Definition 12. A Wigner matrix is a random matrix X =(X i, j ) i, j n where (1) X i, j, i < j are i.i.d (real or complex valued). (2) X i,i, i n are
More informationFINITE-DIMENSIONAL LINEAR ALGEBRA
DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H ROSEN FINITE-DIMENSIONAL LINEAR ALGEBRA Mark S Gockenbach Michigan Technological University Houghton, USA CRC Press Taylor & Francis Croup
More informationNumerical Methods in Matrix Computations
Ake Bjorck Numerical Methods in Matrix Computations Springer Contents 1 Direct Methods for Linear Systems 1 1.1 Elements of Matrix Theory 1 1.1.1 Matrix Algebra 2 1.1.2 Vector Spaces 6 1.1.3 Submatrices
More informationAn Invitation to Modern Number Theory. Steven J. Miller and Ramin Takloo-Bighash PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD
An Invitation to Modern Number Theory Steven J. Miller and Ramin Takloo-Bighash PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD Contents Foreword Preface Notation xi xiii xix PART 1. BASIC NUMBER THEORY
More informationGATE Engineering Mathematics SAMPLE STUDY MATERIAL. Postal Correspondence Course GATE. Engineering. Mathematics GATE ENGINEERING MATHEMATICS
SAMPLE STUDY MATERIAL Postal Correspondence Course GATE Engineering Mathematics GATE ENGINEERING MATHEMATICS ENGINEERING MATHEMATICS GATE Syllabus CIVIL ENGINEERING CE CHEMICAL ENGINEERING CH MECHANICAL
More informationRandom Matrices: Invertibility, Structure, and Applications
Random Matrices: Invertibility, Structure, and Applications Roman Vershynin University of Michigan Colloquium, October 11, 2011 Roman Vershynin (University of Michigan) Random Matrices Colloquium 1 / 37
More informationNumerical Analysis for Statisticians
Kenneth Lange Numerical Analysis for Statisticians Springer Contents Preface v 1 Recurrence Relations 1 1.1 Introduction 1 1.2 Binomial CoefRcients 1 1.3 Number of Partitions of a Set 2 1.4 Horner's Method
More informationThe Essentials of Linear State-Space Systems
:or-' The Essentials of Linear State-Space Systems J. Dwight Aplevich GIFT OF THE ASIA FOUNDATION NOT FOR RE-SALE John Wiley & Sons, Inc New York Chichester Weinheim OAI HOC OUOC GIA HA N^l TRUNGTAMTHANCTINTHUVIIN
More informationMATRIX AND LINEAR ALGEBR A Aided with MATLAB
Second Edition (Revised) MATRIX AND LINEAR ALGEBR A Aided with MATLAB Kanti Bhushan Datta Matrix and Linear Algebra Aided with MATLAB Second Edition KANTI BHUSHAN DATTA Former Professor Department of Electrical
More information!"#$%&'(&)*$%&+",#$$-$%&+./#-+ (&)*$%&+%"-$+0!#1%&
!"#$%&'(&)*$%&",#$$-$%&./#- (&)*$%&%"-$0!#1%&23 44444444444444444444444444444444444444444444444444444444444444444444 &53.67689:5;978?58"@A9;8=B!=89C7DE,6=8FG=CD=CF(76F9C7D!)#!/($"%*$H!I"%"&1/%/.!"JK$&3
More informationRandom matrices: A Survey. Van H. Vu. Department of Mathematics Rutgers University
Random matrices: A Survey Van H. Vu Department of Mathematics Rutgers University Basic models of random matrices Let ξ be a real or complex-valued random variable with mean 0 and variance 1. Examples.
More informationPreface. 2 Linear Equations and Eigenvalue Problem 22
Contents Preface xv 1 Errors in Computation 1 1.1 Introduction 1 1.2 Floating Point Representation of Number 1 1.3 Binary Numbers 2 1.3.1 Binary number representation in computer 3 1.4 Significant Digits
More informationRandom Matrices and Wireless Communications
Random Matrices and Wireless Communications Jamie Evans Centre for Ultra-Broadband Information Networks (CUBIN) Department of Electrical and Electronic Engineering University of Melbourne 3.5 1 3 0.8 2.5
More informationOperator norm convergence for sequence of matrices and application to QIT
Operator norm convergence for sequence of matrices and application to QIT Benoît Collins University of Ottawa & AIMR, Tohoku University Cambridge, INI, October 15, 2013 Overview Overview Plan: 1. Norm
More informationEstimation of large dimensional sparse covariance matrices
Estimation of large dimensional sparse covariance matrices Department of Statistics UC, Berkeley May 5, 2009 Sample covariance matrix and its eigenvalues Data: n p matrix X n (independent identically distributed)
More informationELEMENTARY MATRIX ALGEBRA
ELEMENTARY MATRIX ALGEBRA Third Edition FRANZ E. HOHN DOVER PUBLICATIONS, INC. Mineola, New York CONTENTS CHAPTER \ Introduction to Matrix Algebra 1.1 Matrices 1 1.2 Equality of Matrices 2 13 Addition
More informationRandom regular digraphs: singularity and spectrum
Random regular digraphs: singularity and spectrum Nick Cook, UCLA Probability Seminar, Stanford University November 2, 2015 Universality Circular law Singularity probability Talk outline 1 Universality
More informationLecture 7 MIMO Communica2ons
Wireless Communications Lecture 7 MIMO Communica2ons Prof. Chun-Hung Liu Dept. of Electrical and Computer Engineering National Chiao Tung University Fall 2014 1 Outline MIMO Communications (Chapter 10
More informationxvi xxiii xxvi Construction of the Real Line 2 Is Every Real Number Rational? 3 Problems Algebra of the Real Numbers 7
About the Author v Preface to the Instructor xvi WileyPLUS xxii Acknowledgments xxiii Preface to the Student xxvi 1 The Real Numbers 1 1.1 The Real Line 2 Construction of the Real Line 2 Is Every Real
More informationMathematics Syllabus UNIT I ALGEBRA : 1. SETS, RELATIONS AND FUNCTIONS
Mathematics Syllabus UNIT I ALGEBRA : 1. SETS, RELATIONS AND FUNCTIONS (i) Sets and their Representations: Finite and Infinite sets; Empty set; Equal sets; Subsets; Power set; Universal set; Venn Diagrams;
More informationEigenvalues and Singular Values of Random Matrices: A Tutorial Introduction
Random Matrix Theory and its applications to Statistics and Wireless Communications Eigenvalues and Singular Values of Random Matrices: A Tutorial Introduction Sergio Verdú Princeton University National
More information12.4 Known Channel (Water-Filling Solution)
ECEn 665: Antennas and Propagation for Wireless Communications 54 2.4 Known Channel (Water-Filling Solution) The channel scenarios we have looed at above represent special cases for which the capacity
More informationConceptual Questions for Review
Conceptual Questions for Review Chapter 1 1.1 Which vectors are linear combinations of v = (3, 1) and w = (4, 3)? 1.2 Compare the dot product of v = (3, 1) and w = (4, 3) to the product of their lengths.
More informationSpectral inequalities and equalities involving products of matrices
Spectral inequalities and equalities involving products of matrices Chi-Kwong Li 1 Department of Mathematics, College of William & Mary, Williamsburg, Virginia 23187 (ckli@math.wm.edu) Yiu-Tung Poon Department
More informationApplied Linear Algebra
Applied Linear Algebra Peter J. Olver School of Mathematics University of Minnesota Minneapolis, MN 55455 olver@math.umn.edu http://www.math.umn.edu/ olver Chehrzad Shakiban Department of Mathematics University
More informationInverse Eigenvalue Problems: Theory, Algorithms, and Applications
Inverse Eigenvalue Problems: Theory, Algorithms, and Applications Moody T. Chu North Carolina State University Gene H. Golub Stanford University OXPORD UNIVERSITY PRESS List of Acronyms List of Figures
More informationReview of Some Concepts from Linear Algebra: Part 2
Review of Some Concepts from Linear Algebra: Part 2 Department of Mathematics Boise State University January 16, 2019 Math 566 Linear Algebra Review: Part 2 January 16, 2019 1 / 22 Vector spaces A set
More informationA new method to bound rate of convergence
A new method to bound rate of convergence Arup Bose Indian Statistical Institute, Kolkata, abose@isical.ac.in Sourav Chatterjee University of California, Berkeley Empirical Spectral Distribution Distribution
More information1 Singular Value Decomposition and Principal Component
Singular Value Decomposition and Principal Component Analysis In these lectures we discuss the SVD and the PCA, two of the most widely used tools in machine learning. Principal Component Analysis (PCA)
More informationLessons in Estimation Theory for Signal Processing, Communications, and Control
Lessons in Estimation Theory for Signal Processing, Communications, and Control Jerry M. Mendel Department of Electrical Engineering University of Southern California Los Angeles, California PRENTICE HALL
More informationGEOPHYSICAL INVERSE THEORY AND REGULARIZATION PROBLEMS
Methods in Geochemistry and Geophysics, 36 GEOPHYSICAL INVERSE THEORY AND REGULARIZATION PROBLEMS Michael S. ZHDANOV University of Utah Salt Lake City UTAH, U.S.A. 2OO2 ELSEVIER Amsterdam - Boston - London
More informationMeasure, Integration & Real Analysis
v Measure, Integration & Real Analysis preliminary edition 10 August 2018 Sheldon Axler Dedicated to Paul Halmos, Don Sarason, and Allen Shields, the three mathematicians who most helped me become a mathematician.
More informationLinear Algebra: Characteristic Value Problem
Linear Algebra: Characteristic Value Problem . The Characteristic Value Problem Let < be the set of real numbers and { be the set of complex numbers. Given an n n real matrix A; does there exist a number
More informationMatrix Algorithms. Volume II: Eigensystems. G. W. Stewart H1HJ1L. University of Maryland College Park, Maryland
Matrix Algorithms Volume II: Eigensystems G. W. Stewart University of Maryland College Park, Maryland H1HJ1L Society for Industrial and Applied Mathematics Philadelphia CONTENTS Algorithms Preface xv xvii
More information1 Intro to RMT (Gene)
M705 Spring 2013 Summary for Week 2 1 Intro to RMT (Gene) (Also see the Anderson - Guionnet - Zeitouni book, pp.6-11(?) ) We start with two independent families of R.V.s, {Z i,j } 1 i
More informationPreface to Second Edition... vii. Preface to First Edition...
Contents Preface to Second Edition..................................... vii Preface to First Edition....................................... ix Part I Linear Algebra 1 Basic Vector/Matrix Structure and
More informationELEC E7210: Communication Theory. Lecture 10: MIMO systems
ELEC E7210: Communication Theory Lecture 10: MIMO systems Matrix Definitions, Operations, and Properties (1) NxM matrix a rectangular array of elements a A. an 11 1....... a a 1M. NM B D C E ermitian transpose
More informationNon white sample covariance matrices.
Non white sample covariance matrices. S. Péché, Université Grenoble 1, joint work with O. Ledoit, Uni. Zurich 17-21/05/2010, Université Marne la Vallée Workshop Probability and Geometry in High Dimensions
More informationOPTIMAL CONTROL AND ESTIMATION
OPTIMAL CONTROL AND ESTIMATION Robert F. Stengel Department of Mechanical and Aerospace Engineering Princeton University, Princeton, New Jersey DOVER PUBLICATIONS, INC. New York CONTENTS 1. INTRODUCTION
More informationMultivariate Statistical Analysis
Multivariate Statistical Analysis Fall 2011 C. L. Williams, Ph.D. Lecture 4 for Applied Multivariate Analysis Outline 1 Eigen values and eigen vectors Characteristic equation Some properties of eigendecompositions
More informationAssessing the dependence of high-dimensional time series via sample autocovariances and correlations
Assessing the dependence of high-dimensional time series via sample autocovariances and correlations Johannes Heiny University of Aarhus Joint work with Thomas Mikosch (Copenhagen), Richard Davis (Columbia),
More informationABSTRACT ALGEBRA WITH APPLICATIONS
ABSTRACT ALGEBRA WITH APPLICATIONS IN TWO VOLUMES VOLUME I VECTOR SPACES AND GROUPS KARLHEINZ SPINDLER Darmstadt, Germany Marcel Dekker, Inc. New York Basel Hong Kong Contents f Volume I Preface v VECTOR
More informationA note on a Marčenko-Pastur type theorem for time series. Jianfeng. Yao
A note on a Marčenko-Pastur type theorem for time series Jianfeng Yao Workshop on High-dimensional statistics The University of Hong Kong, October 2011 Overview 1 High-dimensional data and the sample covariance
More informationAPPLIED NUMERICAL LINEAR ALGEBRA
APPLIED NUMERICAL LINEAR ALGEBRA James W. Demmel University of California Berkeley, California Society for Industrial and Applied Mathematics Philadelphia Contents Preface 1 Introduction 1 1.1 Basic Notation
More informationExtreme eigenvalues of Erdős-Rényi random graphs
Extreme eigenvalues of Erdős-Rényi random graphs Florent Benaych-Georges j.w.w. Charles Bordenave and Antti Knowles MAP5, Université Paris Descartes May 18, 2018 IPAM UCLA Inhomogeneous Erdős-Rényi random
More informationMATHEMATICS FOR ECONOMISTS. An Introductory Textbook. Third Edition. Malcolm Pemberton and Nicholas Rau. UNIVERSITY OF TORONTO PRESS Toronto Buffalo
MATHEMATICS FOR ECONOMISTS An Introductory Textbook Third Edition Malcolm Pemberton and Nicholas Rau UNIVERSITY OF TORONTO PRESS Toronto Buffalo Contents Preface Dependence of Chapters Answers and Solutions
More informationIntroduction to Infinite Dimensional Stochastic Analysis
Introduction to Infinite Dimensional Stochastic Analysis By Zhi yuan Huang Department of Mathematics, Huazhong University of Science and Technology, Wuhan P. R. China and Jia an Yan Institute of Applied
More informationJacobians of Matrix Transformations and Functions of Matrix Argument
Jacobians of Matrix Transformations and Functions of Matrix Argument A. M. Mathai Department of Mathematics & Statistics, McGill University World Scientific Singapore *New Jersey London Hong Kong Contents
More informationRandom Matrix Theory Lecture 3 Free Probability Theory. Symeon Chatzinotas March 4, 2013 Luxembourg
Random Matrix Theory Lecture 3 Free Probability Theory Symeon Chatzinotas March 4, 2013 Luxembourg Outline 1. Free Probability Theory 1. Definitions 2. Asymptotically free matrices 3. R-transform 4. Additive
More informationNORMS ON SPACE OF MATRICES
NORMS ON SPACE OF MATRICES. Operator Norms on Space of linear maps Let A be an n n real matrix and x 0 be a vector in R n. We would like to use the Picard iteration method to solve for the following system
More informationChannel capacity estimation using free probability theory
Channel capacity estimation using free probability theory January 008 Problem at hand The capacity per receiving antenna of a channel with n m channel matrix H and signal to noise ratio ρ = 1 σ is given
More informationLecture 7: Positive Semidefinite Matrices
Lecture 7: Positive Semidefinite Matrices Rajat Mittal IIT Kanpur The main aim of this lecture note is to prepare your background for semidefinite programming. We have already seen some linear algebra.
More informationChapter 3 Transformations
Chapter 3 Transformations An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Linear Transformations A function is called a linear transformation if 1. for every and 2. for every If we fix the bases
More informationComprehensive Introduction to Linear Algebra
Comprehensive Introduction to Linear Algebra WEB VERSION Joel G Broida S Gill Williamson N = a 11 a 12 a 1n a 21 a 22 a 2n C = a 11 a 12 a 1n a 21 a 22 a 2n a m1 a m2 a mn a m1 a m2 a mn Comprehensive
More informationLinear Algebra Massoud Malek
CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product
More informationarxiv: v5 [math.na] 16 Nov 2017
RANDOM PERTURBATION OF LOW RANK MATRICES: IMPROVING CLASSICAL BOUNDS arxiv:3.657v5 [math.na] 6 Nov 07 SEAN O ROURKE, VAN VU, AND KE WANG Abstract. Matrix perturbation inequalities, such as Weyl s theorem
More informationNumerical Methods for Engineers. and Scientists. Applications using MATLAB. An Introduction with. Vish- Subramaniam. Third Edition. Amos Gilat.
Numerical Methods for Engineers An Introduction with and Scientists Applications using MATLAB Third Edition Amos Gilat Vish- Subramaniam Department of Mechanical Engineering The Ohio State University Wiley
More informationforms Christopher Engström November 14, 2014 MAA704: Matrix factorization and canonical forms Matrix properties Matrix factorization Canonical forms
Christopher Engström November 14, 2014 Hermitian LU QR echelon Contents of todays lecture Some interesting / useful / important of matrices Hermitian LU QR echelon Rewriting a as a product of several matrices.
More informationRandom Toeplitz Matrices
Arnab Sen University of Minnesota Conference on Limits Theorems in Probability, IISc January 11, 2013 Joint work with Bálint Virág What are Toeplitz matrices? a0 a 1 a 2... a1 a0 a 1... a2 a1 a0... a (n
More informationAn Introduction to Probability Theory and Its Applications
An Introduction to Probability Theory and Its Applications WILLIAM FELLER (1906-1970) Eugene Higgins Professor of Mathematics Princeton University VOLUME II SECOND EDITION JOHN WILEY & SONS Contents I
More informationFluctuations from the Semicircle Law Lecture 4
Fluctuations from the Semicircle Law Lecture 4 Ioana Dumitriu University of Washington Women and Math, IAS 2014 May 23, 2014 Ioana Dumitriu (UW) Fluctuations from the Semicircle Law Lecture 4 May 23, 2014
More information4y Springer NONLINEAR INTEGER PROGRAMMING
NONLINEAR INTEGER PROGRAMMING DUAN LI Department of Systems Engineering and Engineering Management The Chinese University of Hong Kong Shatin, N. T. Hong Kong XIAOLING SUN Department of Mathematics Shanghai
More informationChap 3. Linear Algebra
Chap 3. Linear Algebra Outlines 1. Introduction 2. Basis, Representation, and Orthonormalization 3. Linear Algebraic Equations 4. Similarity Transformation 5. Diagonal Form and Jordan Form 6. Functions
More informationContents. Set Theory. Functions and its Applications CHAPTER 1 CHAPTER 2. Preface... (v)
(vii) Preface... (v) CHAPTER 1 Set Theory Definition of Set... 1 Roster, Tabular or Enumeration Form... 1 Set builder Form... 2 Union of Set... 5 Intersection of Sets... 9 Distributive Laws of Unions and
More informationAlgebraic Curves and Riemann Surfaces
Algebraic Curves and Riemann Surfaces Rick Miranda Graduate Studies in Mathematics Volume 5 If American Mathematical Society Contents Preface xix Chapter I. Riemann Surfaces: Basic Definitions 1 1. Complex
More informationContents. 1 Preliminaries 3. Martingales
Table of Preface PART I THE FUNDAMENTAL PRINCIPLES page xv 1 Preliminaries 3 2 Martingales 9 2.1 Martingales and examples 9 2.2 Stopping times 12 2.3 The maximum inequality 13 2.4 Doob s inequality 14
More informationApplications and fundamental results on random Vandermon
Applications and fundamental results on random Vandermonde matrices May 2008 Some important concepts from classical probability Random variables are functions (i.e. they commute w.r.t. multiplication)
More informationMathematics for Engineers and Scientists
Mathematics for Engineers and Scientists Fourth edition ALAN JEFFREY University of Newcastle-upon-Tyne B CHAPMAN & HALL University and Professional Division London New York Tokyo Melbourne Madras Contents
More informationContents. Acknowledgments
Table of Preface Acknowledgments Notation page xii xx xxi 1 Signals and systems 1 1.1 Continuous and discrete signals 1 1.2 Unit step and nascent delta functions 4 1.3 Relationship between complex exponentials
More informationIrr. Statistical Methods in Experimental Physics. 2nd Edition. Frederick James. World Scientific. CERN, Switzerland
Frederick James CERN, Switzerland Statistical Methods in Experimental Physics 2nd Edition r i Irr 1- r ri Ibn World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI CONTENTS
More informationContents. 2 Sequences and Series Approximation by Rational Numbers Sequences Basics on Sequences...
Contents 1 Real Numbers: The Basics... 1 1.1 Notation... 1 1.2 Natural Numbers... 4 1.3 Integers... 5 1.4 Fractions and Rational Numbers... 10 1.4.1 Introduction... 10 1.4.2 Powers and Radicals of Rational
More informationMath 307 Learning Goals
Math 307 Learning Goals May 14, 2018 Chapter 1 Linear Equations 1.1 Solving Linear Equations Write a system of linear equations using matrix notation. Use Gaussian elimination to bring a system of linear
More informationOrthogonal Polynomials on the Unit Circle
American Mathematical Society Colloquium Publications Volume 54, Part 2 Orthogonal Polynomials on the Unit Circle Part 2: Spectral Theory Barry Simon American Mathematical Society Providence, Rhode Island
More informationLecture 02 Linear Algebra Basics
Introduction to Computational Data Analysis CX4240, 2019 Spring Lecture 02 Linear Algebra Basics Chao Zhang College of Computing Georgia Tech These slides are based on slides from Le Song and Andres Mendez-Vazquez.
More informationMath 102, Winter Final Exam Review. Chapter 1. Matrices and Gaussian Elimination
Math 0, Winter 07 Final Exam Review Chapter. Matrices and Gaussian Elimination { x + x =,. Different forms of a system of linear equations. Example: The x + 4x = 4. [ ] [ ] [ ] vector form (or the column
More informationarxiv: v3 [math-ph] 21 Jun 2012
LOCAL MARCHKO-PASTUR LAW AT TH HARD DG OF SAMPL COVARIAC MATRICS CLAUDIO CACCIAPUOTI, AA MALTSV, AD BJAMI SCHLI arxiv:206.730v3 [math-ph] 2 Jun 202 Abstract. Let X be a matrix whose entries are i.i.d.
More informationApproximating sparse binary matrices in the cut-norm
Approximating sparse binary matrices in the cut-norm Noga Alon Abstract The cut-norm A C of a real matrix A = (a ij ) i R,j S is the maximum, over all I R, J S of the quantity i I,j J a ij. We show that
More informationOn singular values distribution of a matrix large auto-covariance in the ultra-dimensional regime. Title
itle On singular values distribution of a matrix large auto-covariance in the ultra-dimensional regime Authors Wang, Q; Yao, JJ Citation Random Matrices: heory and Applications, 205, v. 4, p. article no.
More informationEIGENVECTOR OVERLAPS (AN RMT TALK) J.-Ph. Bouchaud (CFM/Imperial/ENS) (Joint work with Romain Allez, Joel Bun & Marc Potters)
EIGENVECTOR OVERLAPS (AN RMT TALK) J.-Ph. Bouchaud (CFM/Imperial/ENS) (Joint work with Romain Allez, Joel Bun & Marc Potters) Randomly Perturbed Matrices Questions in this talk: How similar are the eigenvectors
More informationLinear Models 1. Isfahan University of Technology Fall Semester, 2014
Linear Models 1 Isfahan University of Technology Fall Semester, 2014 References: [1] G. A. F., Seber and A. J. Lee (2003). Linear Regression Analysis (2nd ed.). Hoboken, NJ: Wiley. [2] A. C. Rencher and
More informationPreface to the Second Edition. Preface to the First Edition
n page v Preface to the Second Edition Preface to the First Edition xiii xvii 1 Background in Linear Algebra 1 1.1 Matrices................................. 1 1.2 Square Matrices and Eigenvalues....................
More informationSemicircle law on short scales and delocalization for Wigner random matrices
Semicircle law on short scales and delocalization for Wigner random matrices László Erdős University of Munich Weizmann Institute, December 2007 Joint work with H.T. Yau (Harvard), B. Schlein (Munich)
More informationNotes on Linear Algebra and Matrix Theory
Massimo Franceschet featuring Enrico Bozzo Scalar product The scalar product (a.k.a. dot product or inner product) of two real vectors x = (x 1,..., x n ) and y = (y 1,..., y n ) is not a vector but a
More information8.1 Concentration inequality for Gaussian random matrix (cont d)
MGMT 69: Topics in High-dimensional Data Analysis Falll 26 Lecture 8: Spectral clustering and Laplacian matrices Lecturer: Jiaming Xu Scribe: Hyun-Ju Oh and Taotao He, October 4, 26 Outline Concentration
More informationIntroduction. Chapter One
Chapter One Introduction The aim of this book is to describe and explain the beautiful mathematical relationships between matrices, moments, orthogonal polynomials, quadrature rules and the Lanczos and
More informationFundamentals of Matrices
Maschinelles Lernen II Fundamentals of Matrices Christoph Sawade/Niels Landwehr/Blaine Nelson Tobias Scheffer Matrix Examples Recap: Data Linear Model: f i x = w i T x Let X = x x n be the data matrix
More informationPreface. Figures Figures appearing in the text were prepared using MATLAB R. For product information, please contact:
Linear algebra forms the basis for much of modern mathematics theoretical, applied, and computational. The purpose of this book is to provide a broad and solid foundation for the study of advanced mathematics.
More informationSpectral law of the sum of random matrices
Spectral law of the sum of random matrices Florent Benaych-Georges benaych@dma.ens.fr May 5, 2005 Abstract The spectral distribution of a matrix is the uniform distribution on its spectrum with multiplicity.
More information