Matrix Algebra for Linear Models

Name: Matrix Algebra for Linear Models
Brand: Wiley
Price: 91.99 EUR
Availability: OnlineOnly

Marvin H. J. Gruber(Autor*in)

Wiley (Verlag)

Erschienen am 2. Dezember 2013

224 Seiten

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978-1-118-80041-6 (ISBN)

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Inhalt

Intro
Matrix Algebra for Linear Models
Copyright
Contents
Preface
Acknowledgments
Part I Basic Ideas about Matrices and Systems of Linear Equations
Section 1 What Matrices Are and Some Basic Operations with Them
1.1 Introduction
1.2 What Are Matrices and Why Are They Interesting to a Statistician?
1.3 Matrix Notation, Addition, and Multiplication
1.4 Summary
Exercises
Section 2 Determinants and Solving a System of Equations
2.1 Introduction
2.2 Definition of and Formulae for Expanding Determinants
2.3 Some Computational Tricks for the Evaluation of Determinants
2.4 Solution to Linear Equations Using Determinants
2.5 Gauss Elimination
2.6 Summary
Exercises
Section 3 The Inverse of a Matrix
3.1 Introduction
3.2 The Adjoint Method of Finding the Inverse of a Matrix
3.3 Using Elementary Row Operations
3.4 Using the Matrix Inverse to Solve a System of Equations
3.5 Partitioned Matrices and Their Inverses
3.6 Finding the Least Square Estimator
3.7 Summary
Exercises
Section 4 Special Matrices and Facts about Matrices That Will Be Used in the Sequel
4.1 Introduction
4.2 Matrices of the Form aIn+bJ n
4.3 Orthogonal Matrices
4.4 Direct Product of Matrices
4.5 An Important Property of Determinants
4.6 The Trace of a Matrix
4.7 Matrix Differentiation
4.8 The Least Square Estimator Again
4.9 Summary
Exercises
Section 5 Vector Spaces
5.1 Introduction
5.2 What Is a Vector Space?
5.3 The Dimension of a Vector Space
5.4 Inner Product Spaces
5.5 Linear Transformations
5.6 Summary
Exercises
Section 6 The Rank of a Matrix and Solutions to Systems of Equations
6.1 Introduction
6.2 The Rank of a Matrix
6.3 Solving Systems of Equations with Coefficient Matrix of Less than Full Rank
6.4 Summary
Exercises
Part II Eigenvalues, the Singular Value Decomposition, and Principal Components
Section 7 Finding the Eigenvalues of a Matrix
7.1 Introduction
7.2 Eigenvalues and Eigenvectors of a Matrix
7.3 Nonnegative Definite Matrices
7.4 Summary
Exercises
Section 8 The Eigenvalues and Eigenvectors of Special Matrices
8.1 Introduction
8.2 Orthogonal, Nonsingular, and Idempotent Matrices
8.3 The Cayley-Hamilton Theorem
8.4 The Relationship between the Trace, the Determinant, and the Eigenvalues of a Matrix
8.5 The Eigenvalues and Eigenvectors of the Kronecker Product of Two Matrices
8.6 The Eigenvalues and the Eigenvectors of a Matrix of the Form aI + bJ
8.7 The Loewner Ordering
8.8 Summary
Exercises
Section 9 The Singular Value Decomposition (SVD)
9.1 Introduction
9.2 The Existence of the SVD
9.3 Uses and Examples of the SVD
9.4 Summary
Exercises
Section 10 Applications of the Singular Value Decomposition
10.1 Introduction
10.2 Reparameterization of a Non-full-Rank Model to a Full-Rank Model
10.3 Principal Components
10.4 The Multicollinearity Problem
10.5 Summary
Exercises
Section 11 Relative Eigenvalues and Generalizations of the Singular Value Decomposition
11.1 Introduction
11.2 Relative Eigenvalues and Eigenvectors
11.3 Generalizations of the Singular Value Decomposition: Overview
11.4 The First Generalization
11.5 The Second Generalization
11.6 Summary
Exercises
Part III Generalized Inverses
Section 12 Basic Ideas about Generalized Inverses
12.1 Introduction
12.2 What Is a Generalized Inverse and How Is One Obtained?
12.3 The Moore-Penrose Inverse
12.4 Summary
Exercises
Section 13 Characterizations of Generalized Inverses Using the Singular Value Decomposition
13.1 Introduction
13.2 Characterization of the Moore-Penrose Inverse
13.3 Generalized Inverses in Terms of the Moore-Penrose Inverse
13.4 Summary
Exercises
Section 14 Least Square and Minimum Norm Generalized Inverses
14.1 Introduction
14.2 Minimum Norm Generalized Inverses
14.3 Least Square Generalized Inverses
14.4 An Extension of Theorem 7.3 to Positive- Semi-definite Matrices
14.5 Summary
Exercises
Section 15 More Representations of Generalized Inverses
15.1 Introduction
15.2 Another Characterization of the Moore-Penrose Inverse
15.3 Still Another Representation of the Generalized Inverse
15.4 The Generalized Inverse of a Partitioned Matrix
15.5 Summary
Exercises
Section 16 Least Square Estimators for Less than Full-Rank Models
16.1 Introduction
16.2 Some Preliminaries
16.3 Obtaining the LS Estimator
16.4 Summary
Exercises
Part IV Quadratic Forms and the Analysis of Variance
Section 17 Quadratic Forms and Their Probability Distributions
17.1 Introduction
17.2 Examples of Quadratic Forms
17.3 The Chi-Square Distribution
17.4 When Does the Quadratic Form of a Random Variable Have a Chi-Square Distribution?
17.5 When Are Two Quadratic Forms with the Chi-Square Distribution Independent?
17.6 Summary
Exercises
Section 18 Analysis of Variance: Regression Models and the One- and Two-Way Classification
18.1 Introduction
18.2 The Full-Rank General Linear Regression Model
18.3 Analysis of Variance: One-Way Classification
18.4 Analysis of Variance: Two-Way Classification
18.5 Summary
Exercises
Section 19 More ANOVA
19.1 Introduction
19.2 The Two-Way Classification with Interaction
19.3 The Two-Way Classification with One Factor Nested
19.4 Summary
Exercises
Section 20 The General Linear Hypothesis
20.1 Introduction
20.2 The Full-Rank Case
20.3 The Non-full-Rank Case
20.4 Contrasts
20.5 Summary
Exercises
Part V Matrix Optimization Problems
Section 21 Unconstrained Optimization Problems
21.1 Introduction
21.2 Unconstrained Optimization Problems
21.3 The Least Square Estimator Again
21.4 Summary
Exercises
Section 22 Constrained Minimization Problems with Linear Constraints
22.1 Introduction
22.2 An Overview of Lagrange Multipliers
22.3 Minimizing a Second-Degree Form with Respect to a Linear Constraint
22.4 The Constrained Least Square Estimator
22.5 Canonical Correlation
22.6 Summary
Exercises
Section 23 The Gauss-Markov Theorem
23.1 Introduction
23.2 The Gauss-Markov Theorem and the Least Square Estimator
23.3 The Modified Gauss-Markov Theorem and the Linear Bayes Estimator
23.4 Summary
Exercises
Section 24 Ridge Regression-Type Estimators
24.1 Introduction
24.2 Minimizing a Second-Degree Form with Respect to a Quadratic Constraint
24.3 The Generalized Ridge Regression Estimators
24.4 The Mean Square Error of the Generalized Ridge Estimator without Averaging over the Prior Distribution
24.5 The Mean Square Error Averaging over the Prior Distribution
24.6 Summary
Exercises
Answers to Selected Exercises
References
Index

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