Matrix Algebra Useful for Statistics

512 Pages·2017·12.07 MB·English

by  Shayle R. Searle

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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