Matrix Algebra - Gentle, James E.
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Matrix 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…mehr

Matrix 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.
  • Produktdetails
  • Springer Texts in Statistics
  • Verlag: Springer / Springer International Publishing / Springer, Berlin
  • Artikelnr. des Verlages: 978-3-319-64866-8
  • 2. Aufl.
  • Seitenzahl: 680
  • Erscheinungstermin: 21. Oktober 2017
  • Englisch
  • Abmessung: 254mm x 178mm x 36mm
  • Gewicht: 1257g
  • ISBN-13: 9783319648668
  • ISBN-10: 3319648667
  • Artikelnr.: 48607852
¿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).
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


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


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


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


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


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


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


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


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


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


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


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



Linear Algebra.- Basic Vector/Matrix Structure and Notation.- Vectors and Vector Spaces.- Basic Properties of Matrices.- Vector/Matrix Derivatives and Integrals.- Matrix Transformations and Factorizations.- Solution of Linear Systems.- Evaluation of Eigenvalues and Eigenvectors.- Applications in Data Analysis.- Special Matrices and Operations Useful in Modeling and Data Analysis.- Selected Applications in Statistics.- Numerical Methods and Software.- Numerical Methods.- Numerical Linear Algebra.- Software for Numerical Linear Algebra.
"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)

From the reviews:

"[T]his well-written book on matrix algebra reminds me of many classics in the field. It is both concise and timely, and provides a good collection of overviews and reviews of important tools used in statistical methods. The book seems best suited as a supplementary text for various courses in multivariate statistical analysis or linear models. I also can safely recommend this book as a handy resource manual for researchers as well as practitioners."

(Technometrics, May 2008, Vol. 50, No. 2)

"Readership: Students of a course in matrix algebra for statistics, or in statistical computing. ... Recently, quite a number of books on matrices related to statistics have been published ... . computational orientation of this book is probably the main difference between it and these other books. ... I never thought that one could write a matrix book with statistical applications without having C.R. Rao in the references; here the book now is. ... an extensive, personal, and easy-to-read matrix book of high quality. Recommended."

(Simo Puntanen, International Statistical Review, Vol. 75 (3), 2007)

"This book is a remarkable and in a way unusual approach to integrate the two mega fields by a kind of interrelated guide. ... Remarkably the referencing is done by pages ... and the pages are precisely on target, which is proof of the careful writing and editing. ... The book is a careful and interesting exposition of almost encyclopedic coverage of the interrelatedness of matrices and computation, emphasizing also statistical applications. ... a strong, highly recommendable guide to the intricacies of matrices in statistics."

(Götz Uebe, Advances in Statistical Analysis, Vol. 92 (3), 2008)

"This is a very refreshing book covering matrix theory and its applications in statistics and numerical analysis. It has the character of a handbook and is lucidly written. ... A 14 page bibliography that is sufficient to trace the omitted proof details rounds out this book into almost a handbook of current state of the art knowledge in matrix theory and applications. There are eleven sets of exercises and detailed hints and partial solutions also."

(Frank Uhlig, Zentralblatt MATH, Vol. 1133 (11), 2008)

"This book could serve as a text for a course in matrices for statistics ... or, more generally, a course in statistical computing or linear models. ... this can be a useful reference book for such a course or, more generally, as a reference for any statistician who uses matrix algebra extensively. ... Overall, I really enjoyed reading Matrix Algebra: Theory, Computations, and Applications in Statistics, and I would recommend it as a nice reference to anyone interested in linear models, particularly its numerical aspects."

(Abhyuday Mandal, Journal of the American Statistical Association, Vol. 103 (484), December, 2008)