Gruber M.H.J. Matrix Algebra for Linear Models

Wiley, 2014. — 392 p. — ISBN: 1118592557, 9781118592557.Comprehensive in scope, Matrix Algebra for Linear Models offers a succinct summary of matrix theory and its related applications to statistics, especially linear models. The book provides a unified presentation of the mathematical properties and statistical applications of matrices in order to define and manipulate data.
Written for theoretical and applied statisticians, the book utilizes multiple numerical examples to illustrate key ideas, methods, and techniques crucial to understanding matrix algebra’s application in linear models. Matrix Algebra for Linear Models expertly balances concepts and methods allowing for a side-by-side presentation of matrix theory and its linear model applications. Including concise summaries on each topic, the book also features:
Methods of deriving results from the properties of eigenvalues and the singular value decomposition.
Solutions to matrix optimization problems for obtaining more efficient biased estimators for parameters in linear regression models.
A section on the generalized singular value decomposition.
Multiple chapter exercises with selected answers to enhance understanding of the presented material.
Matrix Algebra for Linear Models is an ideal textbook for advanced undergraduate and graduate-level courses on statistics, matrices, and linear algebra. The book is also an excellent reference for statisticians, engineers, economists, and readers interested in the linear statistical model.Basic Ideas about Matrices and Systems of Linear Equations
What Matrices are and Some Basic Operations with Them
Determinants and Solving a System of Equations
The Inverse of a Matrix
Special Matrices and Facts about Matrices that will be Used in the Sequel
Vector Spaces
The Rank of a Matrix and Solutions to Systems of Equations
Eigenvalues, the Singular Value Decomposition, and Principal Components
Finding the Eigenvalues of a Matrix
The Eigenvalues and Eigenvectors of Special Matrices
The Singular Value Decomposition (SVD)
Applications of the Singular Value Decomposition
Relative Eigenvalues and Generalizations of the Singular Value Decomposition
Generalized Inverses
Basic Ideas about Generalized Inverses
Characterizations of Generalized Inverses Using the Singular Value Decomposition
Least Square and Minimum Norm Generalized Inverses
More Representations of Generalized Inverses
Least Square Estimators for Less than Full-Rank Models
Quadratic Forms and the Analysis of Variance
Quadratic Forms and their Probability Distributions
Analysis of Variance: Regression Models and the One- and Two-Way Classification
More ANOVA
The General Linear Hypothesis
Matrix Optimization Problems
Unconstrained Optimization Problems
Constrained Minimization Problems with Linear Constraints
The Gauss–Markov Theorem
Ridge Regression-Type Estimators
Exercises
Answers to Selected Exercises