Linear Algebra and Matrix Analysis for Statistics
By Sudipto Banerjee, Anindya Roy
To Be Published December 26th 2013 by Chapman and Hall/CRC – 416 pages
Description
Linear algebra and the study of matrix algorithms have become fundamental to the development of statistical models. Using a vector-space approach, this book provides an understanding of the major concepts that underlie linear algebra and matrix analysis. Each chapter introduces a key topic, such as infinite-dimensional spaces, and provides illustrative examples. The authors examine recent developments in diverse fields such as spatial statistics, machine learning, data mining, and social network analysis. Complete in its coverage and accessible to students without prior knowledge of linear algebra, the text also includes results that are useful for traditional statistical applications.
Reviews
This beautifully written text is unlike any other in statistical science. It starts at the level of a first undergraduate course in linear algebra, and takes the student all the way up to the graduate level, including Hilbert spaces. It is extremely well crafted and proceeds up through that theory at a very good pace. The statistics chapters are added at just the right places to motivate the reader and illustrate the theory. The book is compactly written and mathematically rigorous, yet the style is lively as well as engaging. This elegant, sophisticated work will serve upper level and graduate statistics education well. All and all a book I wish I could have written.
—Jim Zidek, University of British Columbia, Vancouver, Canada
Contents
Basic Operations
Basic definitions and notations
Matrix addition and scalar-matrix multiplication
Matrix multiplication
Partitioned matrices
The "trace" of a square matrix
Some special matrices
Systems of Linear Equations
Introduction
Gaussian elimination
Gauss-Jordan elimination
Elementary matrices
Homogeneous linear systems
The inverse of a matrix
More on Linear Equations
The LU decomposition
Crout’s Algorithm
LU decomposition with row interchanges
The LDU and Cholesky factorizations
Inverse of partitioned matrices
The LDU decomposition for partitioned matrices
The Sherman-Woodbury-Morrison formula
Euclidean Spaces
Introduction
Vector addition and scalar multiplication
Linear spaces and subspaces
Intersection and sum of subspaces
Linear combinations and spans
Four fundamental subspaces
Linear independence
Basis and dimension
The Rank of a Matrix
Rank and nullity of a matrix
Bases for the four fundamental subspaces
Rank and inverse
Rank factorization
The rank-normal form
Rank of a partitioned matrix
Bases for the fundamental subspaces using the rank normal form
Complementary Subspaces
Sum of subspaces
The dimension of the sum of subspaces
Direct sums and complements
Projectors
Orthogonality, Orthogonal Subspaces, and Projections
Inner product, norms, and orthogonality
Row rank = column rank: A proof using orthogonality
Orthogonal projections
Gram-Schmidt orthogonalization
Orthocomplementary subspaces
The fundamental theorem of linear algebra
More on Orthogonality
Orthogonal matrices
The QR decomposition
Orthogonal projection and projector
Orthogonal projector: Alternative derivations
Sum of orthogonal projectors
Orthogonal triangularization
Revisiting Linear Equations
Introduction
Null spaces and the general solution of linear systems
Rank and linear systems
Generalized inverse of a matrix
Generalized inverses and linear systems
The Moore-Penrose inverse
Determinants
Definitions
Some basic properties of determinants
Determinant of products
Computing determinants
The determinant of the transpose of a matrix — revisited
Determinants of partitioned matrices
Cofactors and expansion theorems
The minor and the rank of a matrix
The Cauchy-Binet formula
The Laplace expansion
Eigenvalues and Eigenvectors
Characteristic polynomial and its roots
Spectral decomposition of real symmetric matrices
Spectral decomposition of Hermitian and normal matrices
Further results on eigenvalues
Singular value decomposition
Quadratic Forms
Introduction
Quadratic forms
Matrices in quadratic forms
Positive and nonnegative definite matrices
Congruence and Sylvester’s law of inertia
Nonnegative definite matrices and minors
Extrema of quadratic forms
Simultaneous diagonalization
Matrix and Vector Norms
Matrix norms
Matrix approximation
Principal component analysis
Hilbert Spaces
Orthogonal projection in Hilbert spaces
Some common Hilbert spaces in statistics
Sobolov spaces
Reproducing kernel Hilbert space
References
Exercises appear at the end of each chapter.
Related Subjects
Name: Linear Algebra and Matrix Analysis for Statistics (Hardback) – Chapman and Hall/CRC
Description: By Sudipto Banerjee, Anindya Roy. Linear algebra and the study of matrix algorithms have become fundamental to the development of statistical models. Using a vector-space approach, this book provides an understanding of the major concepts that underlie linear algebra and matrix analysis...
Categories: Algebra, Statistics & Probability, Statistical Theory & Methods
