Matrix Algebra & Its Applications to Statistics & Econometrics

557 Pages·1998·23.95 MB·English

by  C. Radhakrishna Rao

Matrix Algebra and Its Applications
to Statistics and Econometrics
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Matrix Algebra and Its Applications
to Statistics and Econometrics
c.
Radhakrishna Rao
Pennsylvania State University, USA
M. Bhaskara Rao
North Dakota State University, USA
, • World Scientific
Singapore· New Jersey· London· Hong Kong
Published by
World Scientific Publishing Co. Pte. Ltd.
POBox 128, Farrer Road, Singapore 912805
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Library of Congress Cataloging-in-Publication Data
Rao, C. Radhakrishna (Calyampudi Radhakrishna), 1920-
Matrix algebra and its applications to statistics and econometrics / C.
Radhakrishna Rao and M. Bhaskara Rao.
p. cm.
Includes bibliographical references and index.
ISBN 9810232683 (alk. paper)
I. Matrices. 2. Statistics. 3. Econometrics. I. Bhaskara Rao, M.
QA188.R36 1998
512.9'434--dc21 98-5596
CIP
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
First published 1998
Reprinted 2001, 2004
Copyright © 1998 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, nwy not be reproduced in any form or by any means,
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Printed in Singapore by Utopia Press Pte Ltd
To our wives
BHARGAVl (Mrs. C.R. Rao)
JAYASRI (Mrs. M.B. Rao)
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PREFACE
Matrix algebra and matrix computations have become essential pre
requisites for study and research in many branches of science and tech
nology. It is also of interest to know that statistical applications moti
vated new lines of research in matrix algebra, some examples of which
are generalized inverse of matrices, matrix approximations, generaliza
tions of Chebychev and Kantorovich type inequalities, stochastic ma
trices, generalized projectors, Petrie matrices and limits of eigenvalues
of random matrices. The impact of linear algebra on statistics and
econometrics has been so substantial, in fact, that a number of books
devoted entirely to matrix algebra oriented towards applications in these
two subjects are now available. It has also become a common practice
to devote one chapter or a large appendix on matrix calculus in books
on mathematical statistics and econometrics.
Although there is a large number of books devoted to matrix algebra
and matrix computations, most of them are somewhat specialized in
character. Some of them deal with purely mathematical aspects and
do not give any applications. Others discuss applications using limited
matrix theory. We have attempted to bridge the gap between the two
types. We provide a rigorous treatment of matrix theory and discuss
a variety of applications especially in statistics and econometrics. The
book is aimed at different categories of readers: graduate students in
mathematics who wish to study matrix calculus and get acquainted with
applications in other disciplines, graduate students in statistics, psychol
ogy, economics and engineering who wish to concentrate on applications,
and to research workers who wish to know the current developments in
matrix theory for possible applications in other areas.
This book provides a self-contained, updated and unified treatment
of the theory and applications of matrix methods in statistics and econ~
metrics. All the standard results and the current developments, such
as the generalized inverse of matrices, matrix approximations, matrix
vii
viii MATRIX ALGEBRA THEORY AND APPLICATIONS
differential calculus and matrix decompositions, are brought together
to produce a most comprehensive treatise to serve both as a text in
graduate courses and a reference volume for research students and con
sultants.
It has a large number of examples from different applied areas and
numerous results as complements to illustrate the ubiquity of matrix
algebra in scientific and technological investigations.
It has 16 chapters with the following contents. Chapter 1 introduces
the concept of vector spaces in a very general setup. All the mathemat
ical ideas involved are explained and numerous examples are given. Of
special interest is the construction of orthogonal latin squares using con
cepts of vector spaces. Chapter 2 specializes to unitary and Euclidean
spaces, which are vector spaces in which distances and angles between
vectors are defined. They playa special role in applications.
Chapter 3 discusses linear transformations and matrices. The no
tion of a transformation from one vector space to another is introduced
and the operational role of matrices for this purpose is explained. Thus
matrices are introduced in a natural way and the relationship between
transformations and matrices is emphasized throughout the rest of the
book. Chapters 4, 5, 6 and 7 cover all aspects of matrix calculus. Special
mention may be made of theorems on rank of matrices, factorization of
matrices, eigenvalues and eigenvectors, matrix derivatives and projec
tion operators. Chapter 8 is devoted to generalized inverse of matrices,
a new area in matrix algebra which has been found to be a valuable tool
in developing a unified theory of linear models in statistics and econo
metrics. Chapters 9, 10 and 11 discuss special topics in matrix theory
which are useful in solving optimization problems. Of special interest
are inequalities on singular values of matrices and norms of matrices
which have applications in almost all areas of science and technology.
Chapters 12 and 13 are devoted to the use of matrix methods in the
estimation of parameters in univariate and multivariate linear models.
Concepts of quadratic subspaces and new strategies of solving linear
equations are introduced to provide a unified theory and computational
techniques for the estimation of parameters. Some modern develop
ments in regression theory such as total least squares, estimation of
parameters in mixed linear models and minimum norm quadratic es
timation are discussed in detail using matrix methods. Chapter 14
Preface ix
deals with inequalities which are useful in solving problems in statis
tics and econometrics. Chapter 15 is devoted to non-negative matrices
and Perron-Frobenius theorem which are essential for the study of and
research in econometrics, game theory, decision theory and genetics.
Some miscellaneous results not covered in the main themes of previous
chapters are put together in Chapter 16.
It is a pleasure to thank Marina Tempelman for her patience in typing
numerous revisions of the book.
March 1998
C.R. Rao
M.B. Rao