Applied Matrix Algebra in the Statistical Sciences

722 Pages·2013·9.507 MB·English

by  Basilevsky, Alexander

DOVER BOOKS ON
MATHEMATICS
AN INTRODUCTION TO LINEAR ALGEBRA AND
TENSORS, M. A. Akivis and V. V. Goldberg.
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VECTORS, TENSORS AND THE BASIC EQUATIONS OF
FLUID MECHANICS, Rutherford Aris. (0-486-66110-5)
ASYMPTOTIC EXPANSIONS OF INTEGRALS, Norman
Bleistein and Richard A. Handelsman. (0-486-65082-0)
NON-EUCLIDEAN GEOMETRY, Roberto Bonola.
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INTRODUCTION TO PARTIAL DIFFERENTIAL
EQUATIONS, Arne Broman. (0-486-66158-X)
AN INTRODUCTION TO ORDINARY DIFFERENTIAL
EQUATIONS, Earl A. Coddington. (0-486-65942-9)
MATRICES AND LINEAR TRANSFORMATIONS,
Charles G. Cullen. (0-486-66328-0)
INTRODUCTION TO NONLINEAR DIFFERENTIAL AND
INTEGRAL EQUATIONS, H. T. Davis. (0-486-60971-5)
SOME THEORY OF SAMPLING, W. Edwards Deming.
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2
INTRODUCTION TO LINEAR ALGEBRA AND
DIFFERENTIAL EQUATIONS, John W. Dettman.
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LINEAR PROGRAMMING AND ECONOMIC
ANALYSIS, Robert Dorfman, Paul A. Samuelson and Robert
M. Solow. (0-486-65491-5)
p
THEORY OF H SPACES, Peter L. Duren. (0-486-41184-2)
THE THIRTEEN BOOKS OF EUCLID’S ELEMENTS,
translated with an introduction and commentary by Sir
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0-486-60090-4) Three-volume set
CALCULUS OF VARIATIONS WITH APPLICATIONS,
George M. Ewing. (0-486-64856-7)
DIFFERENTIAL FORMS WITH APPLICATIONS TO THE
PHYSICAL SCIENCES, Harley Flanders. (0-486-66169-5)
AN INTRODUCTION TO THE CALCULUS OF
VARIATIONS, Charles Fox. (0-486-65499-0)
FOUNDATIONS OF MODERN ANALYSIS, Avner
Friedman. (0-486-64062-0)
TECHNICAL CALCULUS WITH ANALYTIC
GEOMETRY, Judith L. Gersting. (0-486-67343-X)
INTRODUCTION TO DIFFERENCE EQUATIONS, Samuel
Goldberg. (0-486-65084-7)
PROBABILITY: An Introduction, Samuel Goldberg.
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DIFFERENTIAL GEOMETRY, Heinrich W. Guggenheimer.
(0-486-63433-7)
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NUMERICAL METHODS FOR SCIENTISTS AND
ENGINEERS, Richard Hamming. (0-486-65241-6)
PROBABILITY: Elements of the Mathematical Theory, C. R.
Heathcote. (0-486-41149-4)
ORDINARY DIFFERENTIAL EQUATIONS, E. L. Ince.
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LIE ALGEBRAS, Nathan Jacobson. (0-486-63832-4)
GREEK MATHEMATICAL THOUGHT AND THE
ORIGIN OF ALGEBRA, Jacob Klein. (0-486-27289-3)
THEORY AND APPLICATION OF INFINITE SERIES,
Konrad Knopp. (0-486-66165-2)
APPLIED ANALYSIS, Cornelius Lanczos. (0-486-65656-X)
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Applied Matrix Algebra in
the Statistical Sciences
Alexander Basilevsky
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Copyright
Copyright © 1983 by Alexander Basilevsky All Rights
Reserved.
Bibliographical Note
This Dover edition, first published in 2005, is an unabridged
republication of the work originally published by
North-Holland, New York, in 1983.
9780486153377
Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y
11501
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To Florence and Roxan Basilevsky
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Table of Contents
DOVER BOOKS ON MATHEMATICS
Title Page
Copyright Page
Dedication
Preface
Chapter 1 - Vectors
Chapter 2 - Vector Spaces
Chapter 3 - Matrices and Systems of Linear Equations
Chapter 4 - Matrices of Special Type
Chapter 5 - Latent Roots and Latent Vectors
Chapter 6 - Generalized Matrix Inverses
Chapter 7 - Nonnegative and Diagonally Dominant
Matrices
References
Index
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Preface
In recent decades matrix algebra and statistics and probability
have become two of the most important areas of theoretical
and applied mathematics, in terms of university curricula as
well as actual research in the social, geographical, human, and
life sciences. With a growing awareness of the usefulness of
matrices in the statistical sciences has come a very elegant
development of statistical theory, combining the principles of
probability and linear algebra. This has greatly increased our
understanding of the algebraic linear structures and properties
of statistical models. At the same time, however, many of the
more specialized theorems of linear algebra, as well as
particular types of matrices commonly used in statistics, are
usually not discussed in the more theoretical linear algebra
texts, whose main purpose is to give the student a broad
perspective of linear algebraic structures rather than a more
specialized view. Usually matrix results of interest to
statisticians are scattered in the appendices and introductory
chapters of advanced statistical publications or are discussed
in journal articles not readily available to the student and
researcher. This in turn tends to influence university programs
in statistics to place less importance on matrix algebra than it
deserves, with the result that students involved in statistics
and other quantitative programs are often forced to rely on
courses taught in pure mathematics in order to obtain even the
most elementary notions of applied linear algebra. The
present volume is therefore an attempt to collect into a single
text some of the main results of matrix algebra that find wide
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use in both applied and theoretical branches of the statistical
sciences, and provide a bridge between linear algebra and
statistical models. It is hoped that this volume will be useful
to students undertaking undergraduate or postgraduate
degrees in quantitative areas such as statistics, probability,
econometrics, psychometrics, sociometrics, biometrics, the
various life and earth sciences, as well as quantitative
geography and demography, by providing a self-contained
book on applied matrix algebra designed for statistical
application. Also, it is the intention of the author to place at
the disposal of the researcher a handy reference book to be
consulted when the need arises. The material has therefore
been organized in a self-contained manner, beginning with
introductory concepts and leading to the more advanced
topics such as latent roots and vectors, generalized inverses,
and nonnegative matrices. At the end of each chapter is a
section dealing with real-world statistical applications, as well
as exercises, in order to provide the reader with concrete
examples (and motivation) of how (and why) matrices are
useful in the statistical sciences, as well as with an illustration
of the matrix algebra. Whenever possible, proofs of the
theorems have been structured in such a way so as to obviate
the use of abstract algebraic notions. The only mathematical
background that is needed is a good knowledge of high school
mathematics, as well as a first course in statistics, although
certain sections, such as those dealing with quadratic forms,
require differential calculus. The author has also attempted to
relate algebraic results to geometric concepts and diagrams in
order to render the material more understandable from the
intuitive point of view. The main purpose of this book,
however, is to provide the reader with a systematic
development of applied matrix theory, and the author has
included more-or-less complete proofs of the theorems.
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