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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Serie: Wiley Series in Probability and Statistics
- Sprache: Englisch
A thoroughly updated guide to matrix algebra and it uses in statistical analysis and features SAS®, MATLAB®, and R throughout
This Second Edition addresses matrix algebra that is useful in the statistical analysis of data as well as within statistics as a whole. The material is presented in an explanatory style rather than a formal theorem-proof format and is self-contained. Featuring numerous applied illustrations, numerical examples, and exercises, the book has been updated to include the use of SAS, MATLAB, and R for the execution of matrix computations. In addition, André I. Khuri, who has extensive research and teaching experience in the field, joins this new edition as co-author. The Second Edition also:
- Contains new coverage on vector spaces and linear transformations and discusses computational aspects of matrices
- Covers the analysis of balanced linear models using direct products of matrices
- Analyzes multiresponse linear models where several responses can be of interest
- Includes extensive use of SAS, MATLAB, and R throughout
- Contains over 400 examples and exercises to reinforce understanding along with select solutions
- Includes plentiful new illustrations depicting the importance of geometry as well as historical interludes
Matrix Algebra Useful for Statistics, Second Edition is an ideal textbook for advanced undergraduate and first-year graduate level courses in statistics and other related disciplines. The book is also appropriate as a reference for independent readers who use statistics and wish to improve their knowledge of matrix algebra.
THE LATE SHAYLE R. SEARLE, PHD, was professor emeritus of biometry at Cornell University. He was the author of Linear Models for Unbalanced Data and Linear Models and co-author of Generalized, Linear, and Mixed Models, Second Edition, Matrix Algebra for Applied Economics, and Variance Components, all published by Wiley. Dr. Searle received the Alexander von Humboldt Senior Scientist Award, and he was an honorary fellow of the Royal Society of New Zealand.
ANDRÉ I. KHURI, PHD, is Professor Emeritus of Statistics at the University of Florida. He is the author of Advanced Calculus with Applications in Statistics, Second Edition and co-author of Statistical Tests for Mixed Linear Models, all published by Wiley. Dr. Khuri is a member of numerous academic associations, among them the American Statistical Association and the Institute of Mathematical Statistics.
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WILEY SERIES IN PROBABILITY AND STATISTICS
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Matrix Algebra Useful for Statistics
Second Edition
Shayle R. Searle
André I. Khuri
Copyright © 2017 by John Wiley & Sons, Inc. All rights reserved.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada.
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Library of Congress Cataloging-in-Publication Data is available.
ISBN: 978-1-118-93514-9
In Memory of Shayle R. Searle, a Good Friend and Colleague
To My Faithful Wife, Ronnie, and Dedicated Children, Marcus and Roxanne, and Their Families
CONTENTS
PREFACE
PREFACE TO THE FIRST EDITION
INTRODUCTION
ABOUT THE COMPANION WEBSITE
PART I DEFINITIONS, BASIC CONCEPTS, AND MATRIX OPERATIONS
1 Vector Spaces, Subspaces, and Linear Transformations
1.1 Vector Spaces
1.2 Base of a Vector Space
1.3 Linear Transformations
Reference
Exercises
2 Matrix Notation and Terminology
2.1 Plotting of a Matrix
2.2 Vectors and Scalars
2.3 General Notation
Exercises
3 Determinants
3.1 Expansion by Minors
3.2 Formal Definition
3.3 Basic Properties
3.4 Elementary Row Operations
3.5 Examples
3.6 Diagonal Expansion
3.7 The Laplace Expansion
3.8 Sums and Differences of Determinants
3.9 A Graphical Representation of a 3 × 3 Determinant
References
Exercises
Notes
4 Matrix Operations
4.1 The Transpose of a Matrix
4.2 Partitioned Matrices
4.3 The Trace of a Matrix
4.4 Addition
4.5 Scalar Multiplication
4.6 Equality and the Null Matrix
4.7 Multiplication
4.8 The Laws of Algebra
4.9 Contrasts With Scalar Algebra
4.10 Direct Sum of Matrices
4.11 Direct Product of Matrices
4.12 The Inverse of a Matrix
4.13 Rank of a Matrix—Some Preliminary Results
4.14 The Number of LIN Rows and Columns in a Matrix
4.15 Determination of The Rank of a Matrix
4.16 Rank and Inverse Matrices
4.17 Permutation Matrices
4.18 Full-Rank Factorization
References
Exercises
5 Special Matrices
5.1 Symmetric Matrices
5.2 Matrices Having all Elements Equal
5.3 Idempotent Matrices
5.4 Orthogonal Matrices
5.5 Parameterization of Orthogonal Matrices
5.6 Quadratic Forms
5.7 Positive Definite Matrices
References
Exercises
6 Eigenvalues and Eigenvectors
6.1 Derivation of Eigenvalues
6.2 Elementary Properties of Eigenvalues
6.3 Calculating Eigenvectors
6.4 The Similar Canonical Form
6.5 Symmetric Matrices
6.6 Eigenvalues of orthogonal and Idempotent Matrices
6.7 Eigenvalues of Direct Products and Direct Sums of Matrices
6.8 Nonzero Eigenvalues of AB and BA
References
Exercises
Notes
7 Diagonalization of Matrices
7.1 Proving the Diagonability Theorem
7.2 Other Results for Symmetric Matrices
7.3 The Cayley–Hamilton Theorem
7.4 The Singular-Value Decomposition
References
Exercises
8 Generalized Inverses
8.1 The Moore–Penrose Inverse
8.2 Generalized Inverses
8.3 Other Names and Symbols
8.4 Symmetric Matrices
References
Exercises
9 Matrix Calculus
9.1 Matrix Functions
9.2 Iterative Solution of Nonlinear Equations
9.3 Vectors of Differential Operators
9.4 Vec and Vech Operators
9.5 Other Calculus Results
9.6 Matrices With Elements That Are Complex Numbers
9.7 Matrix Inequalities
References
Exercises
Notes
PART II APPLICATIONS OF MATRICES IN STATISTICS
10 Multivariate Distributions and Quadratic Forms
10.1 Variance-Covariance Matrices
10.2 Correlation Matrices
10.3 Matrices of Sums of Squares and Cross-Products
10.4 The Multivariate Normal Distribution
10.5 Quadratic Forms and χ
2
-Distributions
10.6 Computing the Cumulative Distribution Function of a Quadratic Form
References
Exercises
11 Matrix Algebra of Full-Rank Linear Models
11.1 Estimation of β by the Method of Least Squares
11.2 Statistical Properties of the Least-Squares Estimator
11.3 Multiple Correlation Coefficient
11.4 Statistical Properties Under the Normality Assumption
11.5 Analysis of Variance
11.6 The Gauss–Markov Theorem
11.7 Testing Linear Hypotheses
11.8 Fitting Subsets of the x-Variables
11.9 The use of the R(.|.) Notation in Hypothesis Testing
References
Exercises
12 Less-Than-Full-Rank Linear Models
12.1 General Description
12.2 The Normal Equations
12.3 Solving the Normal Equations
12.4 Expected values and variances
12.5 Predicted y-Values
12.6 Estimating the Error Variance
12.7 Partitioning the Total Sum of Squares
12.8 Analysis of Variance
12.9 The R( · | · ) Notation
12.10 Estimable Linear Functions
12.11 Confidence Intervals
12.12 Some Particular Models
12.13 The R( · | ·) Notation (continued)
12.14 Reparameterization to a Full-Rank Model
References
Exercises
13 Analysis of Balanced Linear Models Using Direct Products of Matrices
13.1 General Notation for Balanced Linear Models
13.2 Properties Associated with Balanced Linear Models
13.3 Analysis of Balanced Linear Models
References
Exercises
14 Multiresponse Models
14.1 Multiresponse Estimation of Parameters
14.2 Linear Multiresponse Models
14.3 Lack of Fit of a Linear Multiresponse Model
References
Exercises
PART III MATRIX COMPUTATIONS AND RELATED SOFTWARE
15 SAS/IML
15.1 Getting Started
15.2 Defining a Matrix
15.3 Creating a Matrix
15.4 Matrix Operations
15.5 Explanations of SAS Statements Used Earlier in the Text
References
Exercises
16 Use of MATLAB in Matrix Computations
16.1 Arithmetic Operators
16.2 Mathematical Functions
16.3 Construction of Matrices
16.4 Two- and Three-Dimensional Plots
References
Exercises
17 Use of R in Matrix Computations
17.1 Two- and Three-Dimensional Plots
References
Exercises
APPENDIX SOLUTIONS TO EXERCISES
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
INDEX
EULA
List of Tables
Chapter 4
Table 4.1
Table 4.2
Chapter 9
Table 9.1
Chapter 10
Table 10.1
Table 10.2
Chapter 11
Table 11.1
Table 11.2
Table 11.3
Table 11.4
Table 11.5
Table 11.6
Table 11.7
Table 11.8
Table 11.9
Table 11.10
Table 11.11
Chapter 12
Table 12.1
Table 12.2
Table 12.3
Table 12.4
Table 12.5
Table 12.6
Table 12.8
Table 12.9
Table 12.10
Table 12.11
Table 12.12
Table 12.13
Chapter 13
Table 13.1
Table 13.2
Table 13.3
Table 13.4
Table 13.5
Table 13.6
Table 13.7
Table 13.8
Table 13.9
Chapter 14
Table 14.1
Chapter 15
Table 15.1
Chapter 16
Table 16.1
Table 16.2
Chapter 17
Table 17.1
Guide
Cover
Table of Contents
Preface
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Preface
The primary objective of the second edition is to update the material in the first edition. This is a significant undertaking given that the first edition appeared in 1982. It should be first pointed out that this is more than just an update. It is in fact a major revision of the material affecting not only its presentation, but also its applicability and use by the reader.
The second edition consists of three parts. Part I is comprised of Chapters 1–9, which with the exception of Chapter 1, covers material based on an update of Chapters 1–12 in the first edition. These chapters are preceded by an introductory chapter giving historical perspectives on matrix algebra. Chapter 1 is new. It discusses vector spaces and linear transformations that represent an introduction to matrices. Part II addresses applications of matrices in statistics. It consists of Chapters 10–14. Chapters 10–11 constitute an update of Chapters 13–14 in the first edition. Chapter 12 is similar to Chapter 15 in the first edition. It covers models that are less than full rank. Chapter 13 is entirely new. It discusses the analysis of balanced linear models using direct products of matrices. Chapter 14 is also a new addition that covers multiresponse linear models where several responses can be of interest. Part III is new. It covers computational aspects of matrices and consists of three chapters. Chapter 15 is on the use of SAS/IML, Chapter 16 covers the use of MATLAB, and Chapter 17 discusses the implementation of R in matrix computations. These three chapters are self-contained and provide the reader with the necessary tools to carry out all the computations described in the book. The reader can choose whichever software he/she feels comfortable with. It is also quite easy to learn new computational techniques that can be beneficial.
The second edition displays a large number of figures to illustrate certain computational details. This provides a visual depiction of matrix entities such as the plotting of a matrix and the graphical representation of a determinant. In addition, many examples have been included to provide a better understanding of the material.
A new feature in the second edition is the addition of detailed solutions to all the odd-numbered exercises. The even-numbered solutions will be placed online by the publisher. This can be helpful to the reader who desires to use the book as a source for learning matrix algebra.
As with the first edition, the second edition emphasizes the “bringing to a broad spectrum of readers a knowledge of matrix algebra that is useful in the statistical analysis of data and in statistics in general.” The second edition should therefore appeal to all those who desire to gain a better understanding of matrix algebra and its applications in linear models and multivariate statistics. The computing capability that the reader needs is particularly enhanced by the inclusion of Part III on matrix computations.
I am grateful to my wife Ronnie, my daughter Roxanne, and son Marcus for their support and keeping up with my progress in writing the book over the past 3 years. I am also grateful to Steve Quigley, a former editor with John Wiley & Sons, for having given me the opportunity to revise the first edition. Furthermore, my gratitude goes to Julie Platt, an Editor-in-Chief with the SAS Institute, for allowing me to use the SAS software in the second edition for two consecutive years.
ANDRÉ I. KHURI
Jacksonville, FloridaJanuary 2017
Preface to the First Edition
Algebra is a mathematical shorthand for language, and matrices are a shorthand for algebra. Consequently, a special value of matrices is that they enable many mathematical operations, especially those arising in statistics and the quantitative sciences, to be expressed concisely and with clarity. The algebra of matrices is, of course, in no way new, but its presentation is often so surrounded by the trappings of mathematical generality that assimilation can be difficult for readers who have only limited ability or training in mathematics. Yet many such people nowadays find a knowledge of matrix algebra necessary for their work, especially where statistics and/or computers are involved. It is to these people that I address this book, and for them, I have attempted to keep the mathematical presentation as informal as possible.
The pursuit of knowledge frequently involves collecting data, and those responsible for the collecting must appreciate the need for analyzing their data to recover and interpret the information contained therein. Such people must therefore understand some of the mathematical tools necessary for this analysis, to an extent either that they can carry out their own analysis, or that they can converse with statisticians and mathematicians whose help will otherwise be needed. One of the necessary tools is matrix algebra. It is becoming as necessary to science today as elementary calculus has been for generations. Matrices originated in mathematics more than a century ago, but their broad adaptation to science is relatively recent, prompted by the widespread acceptance of statistical analysis of data, and of computers to do that analysis; both statistics and computing rely heavily on matrix algebra. The purpose of this book is therefore that of bringing to a broad spectrum of readers a knowledge of matrix algebra that is useful in the statistical analysis of data and in statistics generally.
The basic prerequisite for using the book is high school algebra. Differential calculus is used on only a few pages, which can easily be omitted; nothing will be lost insofar as a general understanding of matrix algebra is concerned. Proofs and demonstrations of most of the theory are given, for without them the presentation would be lifeless. But in every chapter the theoretical development is profusely illustrated with elementary numerical examples and with illustrations taken from a variety of applied sciences. And the last three chapters are devoted solely to uses of matrix algebra in statistics, with Chapters 14 and 15 outlining two of the most widely used statistical techniques: regression and linear models.
The mainstream of the book is its first 11 chapters, beginning with one on introductory concepts that includes a discussion of subscript and summation notation. This is followed by four chapters dealing with basic arithmetic, special matrices, determinants and inverses. Chapters 6 and 7 are on rank and canonical forms, 8 and 9 deal with generalized inverses and solving linear equations, 10 is a collection of results on partitioned matrices, and 11 describes eigenvalues and eigenvectors. Background theory for Chapter 11 is collected in an appendix, Chapter 11A, some summaries and miscellaneous topics make up Chapter 12, statistical illustrations constitute Chapter 13, and Chapters 14 and 15 describe regression and linear models. All chapters except the last two end with exercises.
Occasional sections and paragraphs can be omitted at a first reading, especially by those whose experience in mathematics is somewhat limited. These portions of the book are printed in small type and, generally speaking, contain material subsidiary to the main flow of the text—material that may be a little more advanced in mathematical presentation than the general level otherwise maintained.
Chapters, and sections within chapters, are numbered with Arabic numerals 1, 2, 3,… Within-chapter references to sections are by section number, but references across chapters use the decimal system, for example, Section 1.3 is Section 3 of Chapter 1. These numbers are also shown in the running head of each page, for example, [1.3] is found on page 4. Numbered equations are (1), (2),…, within each chapter. Those of one chapter are seldom referred to in another, but when they are, the chapter reference is explicit; otherwise “equation (3)” or more simply “(3)” means the equation numbered (3) in the chapter concerned. Exercises are in unnumbered sections and are referenced by their chapter number; for example, Exercise 6.2 is Exercise 2 at the end of Chapter 6.
I am greatly indebted to George P. H. Styan for his exquisitely thorough readings of two drafts of the manuscript and his extensive and very helpful array of comments. Harold V. Henderson’s numerous suggestions for the final manuscript were equally as helpful. Readers of Matrix Algebra for the Biological Sciences (Wiley, 1966), and students in 15 years of my matrix algebra course at Cornell have also contributed many useful ideas. Particular thanks go to Mrs. Helen Seamon for her superb accuracy on the typewriter, patience, and fantastic attention to detail; such attributes are greatly appreciated.
SHAYLE R. SEARLE
Ithaca, New York May 1982
About the Companion Website
This book is accompanied by a companion website:
www.wiley.com/go/searle/matrixalgebra2e
The website includes:
Solutions to even numbered exercises (for instructors only)
Part IDefinitions, Basic Concepts, and Matrix Operations
This is the first of three parts that make up this book. The purpose of Part I is to familiarize the reader with the basic concepts and results of matrix algebra. It is designed to provide the tools needed for the understanding of a wide variety of topics in statistics where matrices are used, such as linear models and multivariate analysis, among others. Some of these topics will be addressed in Part II. Proofs of several theorems are given as we believe that understanding the development of a proof can in itself contribute to acquiring a greater ability in dealing with certain matrix intricacies that may be encountered in statistics. However, we shall not attempt to turn this part into a matrix theory treatise overladen with theorems and proofs, which can be quite insipid. Instead, emphasis will be placed on providing an appreciation of the theory, but without losing track of the objective of learning matrix algebra, namely acquiring the ability to apply matrix results in statistics. The theoretical development in every chapter is illustrated with numerous examples to motivate the learning of the theory. The material in Part II will demonstrate the effectiveness of using such theory in statistics.
Part I consists of the following nine chapters:
Chapter 1: Vector Spaces, Subspaces, and Linear Transformations.
Matrix algebra had its foundation in simultaneous linear equations which represented a linear transformation from one n-dimensional Euclidean space to another of the same dimension. This idea was later extended to include linear transformations between more general spaces, not necessarily of the same dimension. Such linear transformations gave rise to matrices. An n-dimensional Euclidean space is a special case of a wider concept called a vector space.
Chapter 2: Matrix Notation and Terminology.
In order to understand and work with matrices, it is necessary to be quite familiar with the notation and system of terms used in matrix algebra. This chapter defines matrices as rectangular or square arrays of numbers arranged in rows and columns.
Chapter 3: Determinants.
This chapter introduces determinants and provides a description of their basic properties. Various methods of determinantal expansions are included.
Chapter 4: Matrix Operations.
This chapter covers various aspects of matrix operations such as partitioning of matrices, multiplication, direct sum, and direct products of matrices, the inverse and rank of matrices, and full-rank factorization.
Chapter 5: Special Matrices.
Certain types of matrices are frequently used in statistics, such as symmetric, orthogonal, idempotent, positive definite matrices. This chapter also includes different methods to parameterize orthogonal matrices.
Chapter 6: Eigenvalues and Eigenvectors.
A detailed study is given of the eigenvalues and eigenvectors of square matrices, their properties and actual computation. Eigenvalues of certain special matrices, such as symmetric, orthogonal, and idempotent matrices, are discussed, in addition to those that pertain to direct products and direct sums of matrices.
Chapter 7: Diagonalization of Matrices.
Different methods are given to diagonalize matrices that satisfy certain properties. The Cayley–Hamilton theorem, and the singular-value decomposition of matrices are also covered.
Chapter 8: Generalized Inverses.
The Moore–Penrose inverse and the more general generalized inverses of matrices are discussed. Properties of generalized inverses of symmetric matrices are studied, including the special case of the matrix.
Chapter 9: Matrix Calculus.
Coverage is given of calculus results associated with matrices, such as functions of matrices, infinite series of matrices, vectors of differential operators, quadratic forms, differentiation of matrices, traces, and determinants, in addition to matrices of second-order partial derivatives, and matrix inequalities.
1Vector Spaces, Subspaces, and Linear Transformations
The study of matrices is based on the concept of linear transformations between two vector spaces. It is therefore necessary to define what this concept means in order to understand the setup of a matrix. In this chapter, as well as in the remainder of the book, the set of all real numbers is denoted by R, and its elements are referred to as scalars. The set of all n-tuples of real numbers will be denoted by Rn (n ≥ 1).
1.1 Vector Spaces
This section introduces the reader to ideas that are used extensively in many books on linear and matrix algebra. They involve extensions of the Euclidean geometry which are important in the current mathematical literature and are described here as a convenient introductory reference for the reader. We confine ourselves to real numbers and to vectors whose elements are real numbers.
1.1.1 Euclidean Space
A vector (x0, y0)′ of two elements can be thought of as representing a point in a two-dimensional Euclidean space using the familiar Cartesian x, y coordinates, as in Figure 1.1. Similarly, a vector (x0, y0, z0)′ of three elements can represent a point in a three-dimensional Euclidean space, also shown in Figure 1.1. In general, a vector of n elements can be said to represent a point (an n-tuple) in what is called an n-dimensional Euclidean space. This is a special case of a wider concept called a vector space, which we now define.
Definition 1.1 (Vector Spaces)
