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Zhidong Bai
Second Edition
Spectral Analysis of Large
Matrices
Jack W. Silverstein
Dimensional Random
Zhidong Bai
Jack W. Silverstein
Department of Mathematics
jack@unity.ncsu.edu
Springer New York Dordrecht Heidelberg London
or by similar or dissimilar methodology now known or hereafter developed is forbidden.
to proprietary rights.
Printed on acid-free paper
Springer is part of Springer Science+Business Media (www.springer.com)
All rights reserved. This work may not be translated or copied in whole or in part without the written
10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in
The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are
not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject
connection with any form of information storage and retrieval, electronic adaptation, computer software,
ISSN 0172-7397
permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY
ISBN 978-1-4419-0660-1
© Springer Science+Business Media, LLC 2010
DOI 10.1007/978-1-4419-0661-8
e-ISBN 978-1-4419-0661-8
Library of Congress Control Number: 2009942423
Singapore
stabaizd@nus.edu.sg
School of Mathematics and Statistics
KLAS MOE
Northeast Normal University
5268 Renmin Street
Changchun, Jilin 130024
China
baizd@nenu.edu.cn
&
Department of Statistics and Applied Probability
National University of Singapore
6 Science Drive 2
North Carolina State University
Box 8205
Raleigh, NC 27695-8205
Singapore 117546
This book is dedicated to:
Professor Calyampudi Radhakrishna Rao’s 90th Birthday
Professor Ulf Grenander’s 87th Birthday
Professor Yongquan Yin’s 80th Birthday
and to
My wife, Xicun Dan, my sons
Li and Steve Gang, and grandsons
Yongji, and Yonglin
— Zhidong Bai
My children, Hila and Idan
— Jack W. Silverstein
Preface to the Second Edition
The ongoing developments being made in large dimensional data analysis
continue to generate great interest in rando m matrix theory in both theoret-
ical investigations and applications in many disciplines. This has doubtlessly
contributed to the significant demand for this monograph, resulting in its first
printing being sold out. The authors have received many requests to publish
a second edition of the book.
Since the publication of the first edition in 2006, many new results have
been reported in the litera ture. However, due to limitations in space, we
cannot include all new achievements in the second edition. In accordance with
the needs of statistics a nd signal processing, we have added a new chapter on
the limiting behavior of eigenvectors of large dimensional sa mple covariance
matrices. To illustrate the application of RMT to wireless communications
and statistical finance, we have added a chapter on these areas. Certain new
developments are commented on throughout the book. Some typos and erro rs
found in the first edition have been corrected.
The authors would like to express their appreciation to Ms. L¨u Hong for her
help in the preparation of the second edition. They would also like to thank
Professors Ying- Chang Liang, Zhaoben Fang, Baoxue Zhang, and Shurong
Zheng, and Mr. Jiang Hu, for their valuable comments and suggestions. They
also thank the co py editor, Mr. Hal Heinglein, for his car e ful reading, cor-
rections, and helpful suggestions. The first author would like to acknowledge
the support from grants NSFC 10871036, NUS R-155-000- 079-112, and R-
155-000-096-720.
Changchun, China, and Singapore Zhidong Bai
Cary, North Carolina, USA Jack W. Silver stein
March 2009
vii
Preface to the First Edition
This monograph is an introductory book on the theory of r andom matri-
ces (RMT). The theory dates back to the early development of quantum
mechanics in the 1940s and 1950s. In an attempt to explain the complex or-
ganizational structur e o f heavy nuclei, E. Wigner , Professor of Mathematical
Physics at Princeton University, argued that one should no t compute energy
levels from Schr¨odinger’s equation. Instead, one should imagine the complex
nuclei s ystem as a black b ox described by n × n Hamiltonian matrices with
elements drawn from a probability distribution with only mild constr aints
dictated by symmetry cons iderations. Under these assumptions and a mild
condition imposed on the probability measure in the space of matrices, o ne
finds the joint probability density of the n eigenvalues. Based on this con-
sideration, Wigner established the well-known semicircular law. Since then,
RMT has been developed into a big research area in mathematical physics
and probability. Its rapid development can be seen from the following statis-
tics from the Mathscinet data base under keyword Random Matrix on 10 June
2005 (Table 0.1).
Table 0.1 Publication numbers on RMT in 10 year periods since 1955
1955–1964 1965–1974 1975–1984 1985–1994 1995–2004
23 138 249 635 1205
Modern developments in computer science and computing facilities moti-
vate ever widening applications of RMT to many areas.
In statistics, classical limit theorems have been found to be seriously in-
adequate in aiding in the analysis of very high dimensional data.
In the biological sciences, a DNA sequence can be as long as several billion
strands. In financial research, the number of different sto cks can be as large
as tens of thousands.
In wireless communications, the numbe r of users can be several million.
ix
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