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Linear Algebra and Matrix Analysis for Statistics

By Sudipto Banerjee, Anindya Roy

Chapman and Hall/CRC – 2014 – 568 pages

Series: Chapman & Hall/CRC Texts in Statistical Science

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    978-1-42-009538-8
    June 5th 2014
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Description

Linear Algebra and Matrix Analysis for Statistics offers a gradual exposition to linear algebra without sacrificing the rigor of the subject. It presents both the vector space approach and the canonical forms in matrix theory. The book is as self-contained as possible, assuming no prior knowledge of linear algebra.

The authors first address the rudimentary mechanics of linear systems using Gaussian elimination and the resulting decompositions. They introduce Euclidean vector spaces using less abstract concepts and make connections to systems of linear equations wherever possible. After illustrating the importance of the rank of a matrix, they discuss complementary subspaces, oblique projectors, orthogonality, orthogonal projections and projectors, and orthogonal reduction.

The text then shows how the theoretical concepts developed are handy in analyzing solutions for linear systems. The authors also explain how determinants are useful for characterizing and deriving properties concerning matrices and linear systems. They then cover eigenvalues, eigenvectors, singular value decomposition, Jordan decomposition (including a proof), quadratic forms, and Kronecker and Hadamard products. The book concludes with accessible treatments of advanced topics, such as linear iterative systems, convergence of matrices, more general vector spaces, linear transformations, and Hilbert spaces.

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 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

Matrices, Vectors, and Their 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

Singular Value and Jordan Decompositions

Singular value decomposition (SVD)

The SVD and the four fundamental subspaces

SVD and linear systems

SVD, data compression and principal components

Computing the SVD

The Jordan canonical form

Implications of the Jordan canonical form

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

The Kronecker Product and Related Operations

Bilinear interpolation and the Kronecker product

Basic properties of Kronecker products

Inverses, rank and nonsingularity of Kronecker products

Matrix factorizations for Kronecker products

Eigenvalues and determinant

The vec and commutator operators

Linear systems involving Kronecker products

Sylvester’s equation and the Kronecker sum

The Hadamard product

Linear Iterative Systems, Norms, and Convergence

Linear iterative systems and convergence of matrix powers

Vector norms

Spectral radius and matrix convergence

Matrix norms and the Gerschgorin circles

SVD – revisited

Web page ranking and Markov chains

Iterative algorithms for solving linear equations

Abstract Linear Algebra

General vector spaces

General inner products

Linear transformations, adjoint and rank

The four fundamental subspaces - revisited

Inverses of linear transformations

Linear transformations and matrices

Change of bases, equivalence and similar matrices

Hilbert spaces

References

Exercises appear at the end of each chapter.

Name: Linear Algebra and Matrix Analysis for Statistics (Hardback)Chapman and Hall/CRC 
Description: By Sudipto Banerjee, Anindya Roy. Linear Algebra and Matrix Analysis for Statistics offers a gradual exposition to linear algebra without sacrificing the rigor of the subject. It presents both the vector space approach and the canonical forms in matrix theory. The book is as...
Categories: Algebra, Statistics & Probability, Statistical Theory & Methods