Matrix Algebra: Theory, Computations and Applications in Statistics

ISBN-13: 9783031421433 / Twarda / 2024 / 764 str.

James E. Gentle

Spis treści

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

O autorze

James E. Gentle is University Professor Emeritus at George Mason University. He is a Fellow of the American Statistical Association (ASA) and of the American Association for the Advancement of Science. He has held several national offices in the ASA and has served as 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, Computational Statistics, and Statistical Analysis of Financial Data. He is co-editor-in-chief of Wiley Interdisciplinary Reviews: Computational Statistics.

Blurb

This book 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 previous editions had essential updates and comprehensive coverage on critical topics in mathematics.

This 3^rd edition offers a self-contained description of relevant aspects 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. It also includes discussions of the R software package, with numerous examples and exercises.

Matrix Algebra considers various types of matrices encountered in statistics, such as projection matrices and positive definite matrices, and describes special properties of those matrices; as well as describing various applications of matrix theory in statistics, including linear models, multivariate analysis, and stochastic processes. 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. It covers numerical linear algebra—one of the most important subjects in the field of statistical computing. The content includes greater emphases on R, and extensive coverage of statistical linear models.

Matrix Algebra is ideal for graduate and advanced undergraduate students, or as a supplementary text for courses in linear models or multivariate statistics. It’s also ideal for use in a course in statistical computing, or as a supplementary text for various courses that emphasize computations.