Description
Book SynopsisMatrix algebra is one of the most important areas of mathematics for data analysis and for statistical theory. This much-needed work presents the relevant aspects of the theory of matrix algebra for applications in statistics. It moves on to consider the various types of matrices encountered in statistics, such as projection matrices and positive definite matrices, and describes the special properties of those matrices. Finally, it covers numerical linear algebra, beginning with a discussion of the basics of numerical computations, and following up with accurate and efficient algorithms for factoring matrices, solving linear systems of equations, and extracting eigenvalues and eigenvectors.
Trade Review“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)
Table of ContentsPart 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
