Overview

This textbook for graduate and advanced undergraduate students presents the theory of matrix algebra for statistical applications, explores various types of matrices encountered in statistics, and covers numerical linear algebra. Matrix algebra is one of the most important areas of mathematics in data science and in statistical theory, and the second edition of this very popular textbook provides essential updates and comprehensive coverage on critical topics in mathematics in data science and in statistical theory. Part I offers a self-contained description of relevant aspects of the theory of matrix algebra for applications in statistics. It begins with fundamental concepts of vectors and vector spaces; covers basic algebraic properties of matrices and analytic properties of vectors and matrices in multivariate calculus; and concludes with a discussion on operations on matrices in solutions of linear systems and in eigenanalysis. Part II considers various types of matricesencountered in statistics, such as projection matrices and positive definite matrices, and describes special properties of those matrices; and describes various applications of matrix theory in statistics, including linear models, multivariate analysis, and stochastic processes. Part III covers numerical linear algebra—one of the most important subjects in the field of statistical computing. It begins with a discussion of the basics of numerical computations and goes on to describe accurate and efficient algorithms for factoring matrices, how to solve linear systems of equations, and the extraction of eigenvalues and eigenvectors. Although the book is not tied to any particular software system, it describes and gives examples of the use of modern computer software for numerical linear algebra. This part is essentially self-contained, although it assumes some ability to program in Fortran or C and/or the ability to use R or Matlab. The first two parts of the text are ideal for a course in matrix algebra for statistics students or as a supplementary text for various courses in linear models or multivariate statistics. The third part is ideal for use as a text for a course in statistical computing or as a supplementary text for various courses that emphasize computations. New to this edition • 100 pages of additional material • 30 more exercises—186 exercises overall • Added discussion of vectors and matrices with complex elements • Additional material on statistical applications • Extensive and reader-friendly cross references and index

Full Product Details

Author:   James E. Gentle
Publisher:   Springer International Publishing AG
Imprint:   Springer International Publishing AG
Edition:   2nd ed. 2017
Weight:   1.274kg
ISBN:  

9783319648668

ISBN 10:   3319648667
Pages:   648
Publication Date:   21 October 2017
Audience:   College/higher education ,  Tertiary & Higher Education
Replaced By:   9783031421433
Format:   Paperback
Publisher's Status:   Active
Availability:   Manufactured on demand  
We will order this item for you from a manufactured on demand supplier.

Table of Contents

Part I Linear Algebra 1 Basic Vector/Matrix Structure and Notation 1.1 Vectors 1.2 Arrays 1.3 Matrices 1.4 Representation of Data 2 Vectors and Vector Spaces 2.1 Operations on Vectors 2.1.1 Linear Combinations and Linear Independence 2.1.2 Vector Spaces and Spaces of Vectors 2.1.3 Basis Sets for Vector Spaces 2.1.4 Inner Products 2.1.5 Norms 2.1.6 Normalized Vectors 2.1.7 Metrics and Distances 2.1.8 Orthogonal Vectors and Orthogonal Vector Spaces 2.1.9 The “One Vector” 2.2 Cartesian Coordinates and Geometrical Properties of Vectors 2.2.1 Cartesian Geometry 2.2.2 Projections 2.2.3 Angles between Vectors 2.2.4 Orthogonalization Transformations; Gram-Schmidt . 2.2.5 Orthonormal Basis Sets 2.2.6 Approximation of Vectors 2.2.7 Flats, Affine Spaces, and Hyperplanes 2.2.8 Cones 2.2.9 Cross Products in IR3 2.3 Centered Vectors and Variances and Covariances of Vectors 2.3.1 The Mean and Centered Vectors 2.3.2 The Standard Deviation, the Variance, andScaled Vectors 2.3.3 Covariances and Correlations between Vectors Exercises 3 Basic Properties of Matrices 3.1 Basic Definitions and Notation 3.1.1 Matrix Shaping Operators 3.1.2 Partitioned Matrices 3.1.3 Matrix Addition 3.1.4 Scalar-Valued Operators on Square Matrices:The Trace 3.1.5 Scalar-Valued Operators on Square Matrices:The Determinant 3.2 Multiplication of Matrices and Multiplication ofVectors and Matrices 3.2.1 Matrix Multiplication (Cayley) 3.2.2 Multiplication of Matrices with Special Patterns 3.2.3 Elementary Operations on Matrices 3.2.4 The Trace of a Cayley Product that Is Square 3.2.5 The Determinant of a Cayley Product of Square Matrices 3.2.6 Multiplication of Matrices and Vectors 3.2.7 Outer Products 3.2.8 Bilinear and Quadratic Forms; Definiteness 3.2.9 Anisometric Spaces 3.2.10 Other Kinds of Matrix Multiplication 3.3 Matrix Rank and the Inverse of a Matrix 3.3.1 The Rank of Partitioned Matrices, Products of Matrices, and Sums of Matrices 3.3.2 Full Rank Partitioning 3.3.3 Full Rank Matrices and Matrix Inverses 3.3.4 Full Rank Factorization 3.3.5 Equivalent Matrices 3.3.6 Multiplication by Full Rank Matrices 3.3.7 Gramian Matrices: Products of the Form ATA 3.3.8 A Lower Bound on the Rank of a Matrix Product 3.3.9 Determinants of Inverses 3.3.10 Inverses of Products and Sums of Nonsingular Matrices 3.3.11 Inverses of Matrices with Special Forms 3.3.12 Determining the Rank of a Matrix 3.4 More on Partitioned Square Matrices: The Schur Complement 3.4.1 Inverses of Partitioned Matrices 3.4.2 Determinants of Partitioned Matrices 3.5 Linear Systems of Equations 3.5.1 Solutions of Linear Systems 3.5.2 Null Space: The Orthogonal Complement 3.6 Generalized Inverses 3.6.1 Special Generalized Inverses; The Moore-Penrose Inverse 3.6.2 Generalized Inverses of Products and Sums of Matrices 3.6.3 Generalized Inverses of Partitioned Matrices 3.7 Orthogonality 3.8 Eigenanalysis; Canonical Factorizations 3.8.1 Basic Properties of Eigenvalues and Eigenvectors 3.8.2 The Characteristic Polynomial 3.8.3 The Spectrum 3.8.4 Similarity Transformations 3.8.5 Schur Factorization 3.8.6 Similar Canonical Factorization; Diagonalizable Matrices 3.8.7 Properties of Diagonalizable Matrices 3.8.8 Eigenanalysis of Symmetric Matrices 3.8.9 Positive Definite and Nonnegative Definite Matrices 3.8.10 Generalized Eigenvalues and Eigenvectors 3.8.11 Singular Values and the Singular Value Decomposition (SVD) 3.9 Matrix Norms 3.9.1 Matrix Norms Induced from Vector Norms 3.9.2 The Frobenius Norm — The “Usual” Norm 3.9.3 Other Matrix Norms 3.9.4 Matrix Norm Inequalities 3.9.5 The Spectral Radius 3.9.6 Convergence of a Matrix Power Series 3.10 Approximation of Matrices Exercises 4 Vector/Matrix Derivatives and Integrals 4.1 Basics of Differentiation 4.2 Types of Differentiation 4.2.1 Differentiation with Respect to a Scalar 4.2.2 Differentiation with Respect to a Vector 4.2.3 Differentiation with Respect to a Matrix 4.3 Optimization of Scalar-Valued Functions 4.3.1 Stationary Points of Functions 4.3.2 Newton’s Method 4.3.3 Least Squares 4.3.4 Maximum Likelihood 4.3.5 Optimization of Functions with Constraints <4.3.6 Optimization without Differentiation 4.4 Integration and Expectation: Applications to Probability Distributions 4.4.1 Multidimensional Integrals and Integrals InvolvingVectors and Matrices 4.4.2 Integration Combined with Other Operations 4.4.3 Random Variables and Probability Distributions Exercises 5 Matrix Transformations and Factorizations 5.1 Linear Geometric Transformations 5.1.1 Transformations by Orthogonal Matrices 5.1.2 Rotations 5.1.3 Reflections 5.1.4 Translations; Homogeneous Coordinates 5.2 Householder Transformations (Reflections) 5.3 Givens Transformations (Rotations) 5.4 Factorization of Matrices 5.5 LU and LDU Factorizations 5.6 QR Factorization 5.6.1 Householder Reflections to Form the QR Factorization 5.6.2 Givens Rotations to Form the QR Factorization 5.6.3 Gram-Schmidt Transformations to Form theQR Factorization 5.7 Factorizations of Nonnegative Definite Matrices 5.7.1 Square Roots 5.7.2 Cholesky Factorization 5.7.3 Factorizations of a Gramian Matrix 5.8 Nonnegative Matrix Factorization 5.9 Other Incomplete Factorizations Exercises 6 Solution of Linear Systems 6.1 Condition of Matrices 6.1.1 Condition Number 6.1.2 Improving the Condition Number 6.1.3 Numerical Accuracy 6.2 Direct Methods for Consistent Systems 6.2.1 Gaussian Elimination and Matrix Factorizations 6.2.2 Choice of Direct Method 6.3 Iterative Methods for Consistent Systems 6.3.1 The Gauss-Seidel Method withSuccessive Overrelaxation 6.3.2 Conjugate Gradient Methods for SymmetricPositive Definite Systems 6.3.3 Multigrid Methods 6.4 Iterative Refinement 6.5 Updating a Solution to a Consistent System 6.6 Overdetermined Systems; Least Squares 6.6.1 Least Squares Solution of an Overdetermined System 6.6.2 Least Squares with a Full Rank Coefficient Matrix 6.6.3 Least Squares with a Coefficient MatrixNot of Full Rank 6.6.4 Updating a Least Squares Solution of anOverdetermined System 6.7 Other Solutions of Overdetermined Systems 6.7.1 Solutions that Minimize Other Norms of the Residuals 6.7.2 Regularized Solutions 6.7.3 Minimizing Orthogonal Distances Exercises 7 Evaluation of Eigenvalues and Eigenvectors 7.1 General Computational Methods 7.1.1 Numerical Condition of an Eigenvalue Problem 7.1.2 Eigenvalues from Eigenvectors and Vice Versa 7.1.3 Deflation 7.1.4 Preconditioning 7.1.5 Shifting 7.2 Power Method 7.3 Jacobi Method 7.4 QR Method 7.5 Krylov Methods 7.6 Generalized Eigenvalues 7.7 Singular Value Decomposition Exercises Part II Applications in Data Analysis 8 Special Matrices and Operations Useful in Modeling andData Analysis 8.1 Data Matrices and Association Matrices 8.1.1 Flat Files 8.1.2 Graphs and Other Data Structures 8.1.3 Term-by-Document Matrices 8.1.4 Probability Distribution Models 8.1.5 Derived Association Matrices 8.2 Symmetric Matrices and Other Unitarily Diagonalizable Matrices 8.2.1 Some Important Properties of Symmetric Matrices 8.2.2 Approximation of Symmetric Matrices and an Important Inequality 8.2.3 Normal Matrices 8.3 Nonnegative Definite Matrices; Cholesky Factorization 8.4 Positive Definite Matrices 8.5 Idempotent and Projection Matrices 8.5.1 Idempotent Matrices 8.5.2 Projection Matrices: Symmetric Idempotent Matrices 8.6 Special Matrices Occurring in Data Analysis 8.6.1 Gramian Matrices 8.6.2 Projection and Smoothing Matrices 8.6.3 Centered Matrices and Variance-Covariance Matrices 8.6.4 The Generalized Variance 8.6.5 Similarity Matrices 8.6.6 Dissimilarity Matrices 8.7 Nonnegative and Positive Matrices 8.7.1 Properties of Square Positive Matrices 8.7.2 Irreducible Square Nonnegative Matrices 8.7.3 Stochastic Matrices 8.7.4 Leslie Matrices 8.8 Other Matrices with Special Structures 8.8.1 Helmert Matrices 8.8.2 Vandermonde Matrices 8.8.3 Hadamard Matrices and Orthogonal Arrays 8.8.4 Toeplitz Matrices 8.8.5 Circulant Matrices 8.8.6 Fourier Matrices and the Discrete Fourier Transform 8.8.7 Hankel Matrices 8.8.8 Cauchy Matrices 8.8.9 Matrices Useful in Graph Theory 8.8.10 M-Matrices Exercises 9 Selected Applications in Statistics 9.1 Multivariate Probability Distributions 9.1.1 Basic Definitions and Properties 9.1.2 The Multivariate Normal Distribution 9.1.3 Derived Distributions and Cochran’s Theorem 9.2 Linear Models 9.2.1 Fitting the Model 9.2.2 Linear Models and Least Squares 9.2.3 Statistical Inference 9.2.4 The Normal Equations and the Sweep Operator 9.2.5 Linear Least Squares Subject to LinearEquality Constraints 9.2.6 Weighted Least Squares 9.2.7 Updating Linear Regression Statistics 9.2.8 Linear Smoothing 9.2.9 Multivariate Linear Models 9.3 Principal Components 9.3.1 Principal Components of a Random Vector 9.3.2 Principal Components of Data 9.4 Condition of Models and Data 9.4.1 Ill-Conditioning in Statistical Applications 9.4.2 Variable Selection 9.4.3 Principal Components Regression 9.4.4 Shrinkage Estimation 9.4.5 Statistical Inference about the Rank of a Matrix 9.4.6 Incomplete Data 9.5 Optimal Design 9.6 Multivariate Random Number Generation 9.7 Stochastic Processes 9.7.1 Markov Chains 9.7.2 Markovian Population Models 9.7.3 Autoregressive Processes Exercises Part III Numerical Methods and Software 10 Numerical Methods 10.1 Digital Representation of Numeric Data 10.1.1 The Fixed-Point Number System 10.1.2 The Floating-Point Model for Real Numbers 10.1.3 Language Constructs for Representing Numeric Data 10.1.4 Other Variations in the Representation of Data;Portability of Data 10.2 Computer Operations on Numeric Data 10.2.1 Fixed-Point Operations 10.2.2 Floating-Point Operations 10.2.3 Exact Computations 10.2.4 Language Constructs for Operations onNumeric Data 10.3 Numerical Algorithms and Analysis 10.3.1 Error in Numerical Computations 10.3.2 Efficiency 10.3.3 Iterations and Convergence <10.3.4 Other Computational Techniques Exercises 11 Numerical Linear Algebra 11.1 Computer Representation of Vectors and Matrices 11.2 General Computational Considerations forVectors and Matrices 11.2.1 Relative Magnitudes of Operands 11.2.2 Iterative Methods 11.2.3 Assessing Computational Errors 11.3 Multiplication of Vectors and Matrices 11.4 Other Matrix Computations Exercises 12 Software for Numerical Linear Algebra 12.1 General Considerations 12.2 Libraries 12.2.1 BLAS 12.2.2 Level 2 and Level 3 BLAS and Related Libraries 12.2.3 Libraries for High Performance Computing 12.2.4 Matrix Storage Modes 12.2.5 Language-Specific Libraries 12.2.6 The IMSLTM Libraries 12.3 General Purpose Languages 12.3.1 Programming Considerations 12.3.2 Modern Fortran 12.3.3 C and C++ 12.3.4 Python <12.4 Interactive Systems for Array Manipulation 12.4.1 R 12.4.2 MATLABR and Octave 12.4.3 Other Systems 12.5 Software for Statistical Applications 12.6 Test Data Exercises Appendices and Back Matter A Notation and Definitions A.1 General Notation A.2 Computer Number Systems A.3 General Mathematical Functions and Operators A.4 Linear Spaces and Matrices A.5 Models and Data B Solutions and Hints for Selected Exercises Bibliography Index

Reviews

Gentle has put in a lot of time and effort to writing this book with careful attention to details. ... it is all needed to make sure the student has a firm and solid understanding of matrix algebra on the graduate level. I would recommend this book for all those who teach graduate level matrix algebra or ... to those undergraduate students who wish to have an independent study. (Peter Olszewski, MAA Reviews, January 2018)

Gentle has put in a lot of time and effort to writing this book with careful attention to details. ... it is all needed to make sure the student has a firm and solid understanding of matrix algebra on the graduate level. I would recommend this book for all those who teach graduate level matrix algebra or ... to those undergraduate students who wish to have an independent study. (Peter Olszewski, MAA Reviews, January, 2018) Beautifully written, easy to read, with a well subindexed index of 16 pages and a bibliography of 13 that includes most modern and relevant textbooks and articles in the area of matrix theory and computations, as well as for statistics and big data computations. (Frank Uhlig, zbMATH 1386.15002, 2018) This very reader-friendly written volume presents an opportunity to graduate students and researchers to enjoy reading on the classical matrix analysis in its modern applications to statistics and to implement these methods in practical problem solving. (Stan Lipovetsky, Technometrics, Vol. 60 (2), 2018)

Author Information

​James E. Gentle, PhD, is University Professor of Computational Statistics at George Mason University. He is a Fellow of the American Statistical Association (ASA) and of the American Association for the Advancement of  Science. Professor Gentle has held several national offices in the ASA and has served as editor and associate editor of journals of the ASA as well as for other journals in statistics and computing. He is author of Random Number Generation and Monte Carlo Methods (Springer, 2003) and Computational Statistics (Springer, 2009).

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