A Matrix Handbook for Statisticians

Name: A Matrix Handbook for Statisticians
Brand: Wiley
Price: 148.99 EUR
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George A. F. Seber(Autor*in)

Wiley (Verlag)

Erschienen am 30. Juni 2008

576 Seiten

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A comprehensive, must-have handbook of matrix methods with a uniqueemphasis on statistical applications
This timely book, A Matrix Handbook for Statisticians, providesa comprehensive, encyclopedic treatment of matrices as they relateto both statistical concepts and methodologies. Written by anexperienced authority on matrices and statistical theory, thishandbook is organized by topic rather than mathematicaldevelopments and includes numerous references to both the theorybehind the methods and the applications of the methods. A uniformapproach is applied to each chapter, which contains four parts: adefinition followed by a list of results; a short list ofreferences to related topics in the book; one or more references toproofs; and references to applications. The use of extensivecross-referencing to topics within the book and externalreferencing to proofs allows for definitions to be located easilyas well as interrelationships among subject areas to berecognized.
A Matrix Handbook for Statisticians addresses the need formatrix theory topics to be presented together in one book andfeatures a collection of topics not found elsewhere under onecover. These topics include:
* Complex matrices
* A wide range of special matrices and their properties
* Special products and operators, such as the Kroneckerproduct
* Partitioned and patterned matrices
* Matrix analysis and approximation
* Matrix optimization
* Majorization
* Random vectors and matrices
* Inequalities, such as probabilistic inequalities
Additional topics, such as rank, eigenvalues, determinants,norms, generalized inverses, linear and quadratic equations,differentiation, and Jacobians, are also included. The book assumesa fundamental knowledge of vectors and matrices, maintains areasonable level of abstraction when appropriate, and provides acomprehensive compendium of linear algebra results with use orpotential use in statistics. A Matrix Handbook for Statisticians isan essential, one-of-a-kind book for graduate-level courses inadvanced statistical studies including linear and nonlinear models,multivariate analysis, and statistical computing. It also serves asan excellent self-study guide for statistical researchers.

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Inhalt

Intro
A MATRIX HANDBOOK FOR STATISTICIANS
CONTENTS
Preface
1 Notation
1.1 General Definitions
1.2 Some Continuous Univariate Distributions
1.3 Glossary of Notation
2 Vectors, Vector Spaces, and Convexity
2.1 Vector Spaces
2.1.1 Definitions
2.1.2 Quadratic Subspaces
2.1.3 Sums and Intersections of Subspaces
2.1.4 Span and Basis
2.1.5 Isomorphism
2.2 Inner Products
2.2.1 Definition and Properties
2.2.2 Functionals
2.2.3 Orthogonality
2.2.4 Column and Null Spaces
2.3 Projections
2.3.1 General Projections
2.3.2 Orthogonal Projections
2.4 Metric Spaces
2.5 Convex Sets and Functions
2.6 Coordinate Geometry
2.6.1 Hyperplanes and Lines
2.6.2 Quadratics
2.6.3 Areas and Volumes
3 Rank
3.1 Some General Properties
3.2 Matrix Products
3.3 Matrix Cancellation Rules
3.4 Matrix Sums
3.5 Matrix Differences
3.6 Partitioned and Patterned Matrices
3.7 Maximal and Minimal Ranks
3.8 Matrix Index
4 Matrix Functions: Inverse, Transpose, Trace, Determinant, and Norm
4.1 Inverse
4.2 Transpose
4.3 Trace
4.4 Determinants
4.4.1 Introduction
4.4.2 Adjoint Matrix
4.4.3 Compound Matrix
4.4.4 Expansion of a Determinant
4.5 Permanents
4.6 Norms
4.6.1 Vector Norms
4.6.2 Matrix Norms
4.6.3 Unitarily Invariant Norms
4.6.4 M, N-Invariant Norms
4.6.5 Computational Accuracy
5 Complex, Hermitian, and Related Matrices
5.1 Complex Matrices
5.1.1 Some General Results
5.1.2 Determinants
5.2 Hermitian Matrices
5.3 Skew-Hermitian Matrices
5.4 Complex Symmetric Matrices
5.5 Real Skew-Symmetric Matrices
5.6 Normal Matrices
5.7 Quaternions
6 Eigenvalues, Eigenvectors, and Singular Values
6.1 Introduction and Definitions
6.1.1 Characteristic Polynomial
6.1.2 Eigenvalues
6.1.3 Singular Values
6.1.4 Functions of a Matrix
6.1.5 Eigenvectors
6.1.6 Hermitian Matrices
6.1.7 Computational Methods
6.1.8 Generalized Eigenvalues
6.1.9 Matrix Products
6.2 Variational Characteristics for Hermitian Matrices
6.3 Separation Theorems
6.4 Inequalities for Matrix Sums
6.5 Inequalities for Matrix Differences
6.6 Inequalities for Matrix Products
6.7 Antieigenvalues and Antieigenvectors
7 Generalized Inverses
7.1 Definitions
7.2 Weak Inverses
7.2.1 General Properties
7.2.2 Products of Matrices
7.2.3 Sums and Differences of Matrices
7.2.4 Real Symmetric Matrices
7.2.5 Decomposition Methods
7.3 Other Inverses
7.3.1 Reflexive (g12) Inverse
7.3.2 Minimum Norm (g14) Inverse
7.3.3 Minimum Norm Reflexive (g124) Inverse
7.3.4 Least Squares (g13) Inverse
7.3.5 Least Squares Reflexive (g123) Inverse
7.4 Moore-Penrose (g1234) Inverse
7.4.1 General Properties
7.4.2 Sums of Matrices
7.4.3 Products of Matrices
7.5 Group Inverse
7.6 Some General Properties of Inverses
8 Some Special Matrices
8.1 Orthogonal and Unitary Matrices
8.2 Permutation Matrices
8.3 Circulant, Toeplitz, and Related Matrices
8.3.1 Regular Circulant
8.3.2 Symmetric Regular Circulant
8.3.3 Symmetric Circulant
8.3.4 Toeplitz Matrix
8.3.5 Persymmetric Matrix
8.3.6 Cross-Symmetric (Centrosymmetric) Matrix
8.3.7 Block Circulant
8.3.8 Hankel Matrix
8.4 Diagonally Dominant Matrices
8.5 Hadamard Matrices
8.6 Idempotent Matrices
8.6.1 General Properties
8.6.2 Sums of Idempotent Matrices and Extensions
8.6.3 Products of Idempotent Matrices
8.7 Tripotent Matrices
8.8 Irreducible Matrices
8.9 Triangular Matrices
8.10 Hessenberg Matrices
8.11 Tridiagonal Matrices
8.12 Vandermonde and Fourier Matrices
8.12.1 Vandermonde Matrix
8.12.2 Fourier Matrix
8.13 Zero-One (0,1) Matrices
8.14 Some Miscellaneous Matrices and Arrays
8.14.1 Krylov Matrix
8.14.2 Nilpotent and Unipotent Matrices
8.14.3 Payoff Matrix
8.14.4 Stable and Positive Stable Matrices
8.14.5 P-Matrix
8.14.6 Z- and M-Matrices
8.14.7 Three-Dimensional Arrays
9 Non-Negative Vectors and Matrices
9.1 Introduction
9.1.1 Scaling
9.1.2 Modulus of a Matrix
9.2 Spectral Radius
9.2.1 General Properties
9.2.2 Dominant Eigenvalue
9.3 Canonical Form of a Non-negative Matrix
9.4 Irreducible Matrices
9.4.1 Irreducible Non-negative Matrix
9.4.2 Periodicity
9.4.3 Non-negative and Nonpositive Off-Diagonal Elements
9.4.4 Perron Matrix
9.4.5 Decomposable Matrix
9.5 Leslie Matrix
9.6 Stochastic Matrices
9.6.1 Basic Properties
9.6.2 Finite Homogeneous Markov Chain
9.6.3 Countably Infinite Stochastic Matrix
9.6.4 Infinite Irreducible Stochastic Matrix
9.7 Doubly Stochastic Matrices
10 Positive Definite and Non-negative Definite Matrices
10.1 Introduction
10.2 Non-negative Definite Matrices
10.2.1 Some General Properties
10.2.2 Gram Matrix
10.2.3 Doubly Non-negative Matrix
10.3 Positive Definite Matrices
10.4 Pairs of Matrices
10.4.1 Non-negative or Positive Definite Difference
10.4.2 One or More Non-negative Definite Matrices
11 Special Products and Operators
11.1 Kronecker Product
11.1.1 Two Matrices
11.1.2 More than Two Matrices
11.2 Vec Operator
11.3 Vec-Permutation (Commutation) Matrix
11.4 Generalized Vec-Permutation Matrix
11.5 Vech Operator
11.5.1 Symmetric Matrix
11.5.2 Lower-Triangular Matrix
11.6 Star Operator
11.7 Hadamard Product
11.8 Rao-Khatri Product
12 Inequalities
12.1 Cauchy-Schwarz inequalities
12.1.1 Real Vector Inequalities and Extensions
12.1.2 Complex Vector Inequalities
12.1.3 Real Matrix Inequalities
12.1.4 Complex Matrix Inequalities
12.2 Hölder's Inequality and Extensions
12.3 Minkowski's Inequality and Extensions
12.4 Weighted Means
12.5 Quasilinearization (Representation) Theorems
12.6 Some Geometrical Properties
12.7 Miscellaneous Inequalities
12.7.1 Determinants
12.7.2 Trace
12.7.3 Quadratics
12.7.4 Sums and Products
12.8 Some Identities
13 Linear Equations
13.1 Unknown Vector
13.1.1 Consistency
13.1.2 Solutions
13.1.3 Homogeneous Equations
13.1.4 Restricted Equations
13.2 Unknown Matrix
13.2.1 Consistency
13.2.2 Some Special Cases
14 Partitioned Matrices
14.1 Schur Complement
14.2 Inverses
14.3 Determinants
14.4 Positive and Non-negative Definite Matrices
14.5 Eigenvalues
14.6 Generalized Inverses
14.6.1 Weak Inverses
14.6.2 Moore-Penrose Inverses
14.7 Miscellaneous partitions
15 Patterned Matrices
15.1 Inverses
15.2 Determinants
15.3 Perturbations
15.4 Matrices with Repeated Elements and Blocks
15.5 Generalized Inverses
15.5.1 Weak Inverses
15.5.2 Moore-Penrose Inverses
16 Factorization of Matrices
16.1 Similarity Reductions
16.2 Reduction by Elementary Transformations
16.2.1 Types of Transformation
16.2.2 Equivalence Relation
16.2.3 Echelon Form
16.2.4 Hermite Form
16.3 Singular Value Decomposition (SVD)
16.4 Triangular Factorizations
16.5 Orthogonal-Triangular Reductions
16.6 Further Diagonal or Tridiagonal Reductions
16.7 Congruence
16.8 Simultaneous Reductions
16.9 Polar Decomposition
16.10 Miscellaneous Factorizations
17 Differentiation
17.1 Introduction
17.2 Scalar Differentiation
17.2.1 Differentiation with Respect to t
17.2.2 Differentiation with Respect to a Vector Element
17.2.3 Differentiation with Respect to a Matrix Element
17.3 Vector Differentiation: Scalar Function
17.3.1 Basic Results
17.3.2 x = vec X
17.3.3 Function of a Function
17.4 Vector Differentiation: Vector Function
17.5 Matrix Differentiation: Scalar Function
17.5.1 General Results
17.5.2 f = trace
17.5.3 f = determinant
17.5.4 f = yrs
17.5.5 f = eigenvalue
17.6 Transformation Rules
17.7 Matrix Differentiation: Matrix Function
17.8 Matrix Differentials
17.9 Perturbation Using Differentials
17.10 Matrix Linear Differential Equations
17.11 Second-Order Derivatives
17.12 Vector Difference Equations
18 Jacobians
18.1 Introduction
18.2 Method of Differentials
18.3 Further Techniques
18.3.1 Chain Rule
18.3.2 Exterior (Wedge) Product of Differentials
18.3.3 Induced Functional Equations
18.3.4 Jacobians Involving Transposes
18.3.5 Patterned Matrices and L-Structures
18.4 Vector Transformations
18.5 Jacobians for Complex Vectors and Matrices
18.6 Matrices with Functionally Independent Elements
18.7 Symmetric and Hermitian Matrices
18.8 Skew-Symmetric and Skew-Hermitian Matrices
18.9 Triangular Matrices
18.9.1 Linear Transformations
18.9.2 Nonlinear Transformations of X
18.9.3 Decompositions with One Skew-Symmetric Matrix
18.9.4 Symmetric Y
18.9.5 Positive Definite Y
18.9.6 Hermitian Positive Definite Y
18.9.7 Skew-Symmetric Y
18.9.8 LU Decomposition
18.10 Decompositions Involving Diagonal Matrices
18.10.1 Square Matrices
18.10.2 One Triangular Matrix
18.10.3 Symmetric and Skew-Symmetric Matrices
18.11 Positive Definite Matrices
18.12 Caley Transformation
18.13 Diagonalizable Matrices
18.14 Pairs of Matrices
19 Matrix Limits, Sequences, and Series
19.1 Limits
19.2 Sequences
19.3 Asymptotically Equivalent Sequences
19.4 Series
19.5 Matrix Functions
19.6 Matrix Exponentials
20 Random Vectors
20.1 Notation
20.2 Variances and Covariances
20.3 Correlations
20.3.1 Population Correlations
20.3.2 Sample Correlations
20.4 Quadratics
20.5 Multivariate Normal Distribution
20.5.1 Definition and Properties
20.5.2 Quadratics in Normal Variables
20.5.3 Quadratics and Chi-Squared
20.5.4 Independence and Quadratics
20.5.5 Independence of Several Quadratics
20.6 Complex Random Vectors
20.7 Regression Models
20.7.1 V Is the Identity Matrix
20.7.2 V Is Positive Definite
20.7.3 V Is Non-negative Definite
20.8 Other Multivariate Distributions
20.8.1 Multivariate t-Distribution
20.8.2 Elliptical and Spherical Distributions
20.8.3 Dirichlet Distributions
21 Random Matrices
21.1 Introduction
21.2 Generalized Quadratic Forms
21.2.1 General Results
21.2.2 Wishart Distribution
21.3 Random Samples
21.3.1 One Sample
21.3.2 Two Samples
21.4 Multivariate Linear Model
21.4.1 Least Squares Estimation
21.4.2 Statistical Inference
21.4.3 Two Extensions
21.5 Dimension Reduction Techniques
21.5.1 Principal Component Analysis (PCA)
21.5.2 Discriminant Coordinates
21.5.3 Canonical Correlations and Variates
21.5.4 Latent Variable Methods
21.5.5 Classical (Metric) Scaling
21.6 Procrustes Analysis (Matching Configurations)
21.7 Some Specific Random Matrices
21.8 Allocation Problems
21.9 Matrix-Variate Distributions
21.10 Matrix Ensembles
22 Inequalities for Probabilities and Random Variables
22.1 General Probabilities
22.2 Bonferroni-Type Inequalities
22.3 Distribution-Free Probability Inequalities
22.3.1 Chebyshev-Type Inequalities
22.3.2 Kolmogorov-Type Inequalities
22.3.3 Quadratics and Inequalities
22.4 Data Inequalities
22.5 Inequalities for Expectations
22.6 Multivariate Inequalities
22.6.1 Convex Subsets
22.6.2 Multivariate Normal
22.6.3 Inequalities For Other Distributions
23 Majorization
23.1 General Properties
23.2 Schur Convexity
23.3 Probabilities and Random variables
24 Optimization and Matrix Approximation
24.1 Stationary Values
24.2 Using Convex and Concave Functions
24.3 Two General Methods
24.3.1 Maximum Likelihood
24.3.2 Least Squares
24.4 Optimizing a Function of a Matrix
24.4.1 Trace
24.4.2 Norm
24.4.3 Quadratics
24.5 Optimal Designs
References
Index

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