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Matrix algebra for linear models / Marvin H. J. Gruber
Resource type: Ressourcentyp: Buch (Online)Buch (Online)Sprache: Englisch Verlag: Hoboken, New Jersey : Wiley, [2014]Auflage: Online-AusgBeschreibung: Online-Ressource (1 online resource (1 online resource (xv, 375 pages).))ISBN:- 1118800419
- 9781118608814
- 111860881X
- 9781118800416
- 9781306191272
- 1306191270
- 9781118592557
- 519.5/36 23
- 519.536
- QA279
- QA279 .G78 2013
Inhalte:
Zusammenfassung: Matrix methods have evolved from a tool for expressing statistical problems to an indispensable part of the development, understanding, and use of various types of complex statistical analyses. This evolution has made matrix methods a vital part of statistical education. Traditionally, matrix methods are taught in courses on everything from regression analysis to stochastic processes, thus creating a fractured view of the topic. Matrix Algebra for Linear Models offers readers a unique, unified view of matrix analysis theory (where and when necessary), methods, and their applications. Written fPPN: PPN: 807205583Package identifier: Produktsigel: ZDB-26-MYL | ZDB-30-PAD | ZDB-30-PQE
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 Determinants2.5 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 Matrices4.5 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 ComponentsSection 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 + bJ8.7 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
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