1st Edition

Linear Algebra and Matrix Analysis for Statistics

By Sudipto Banerjee, Anindya Roy Copyright 2014
    582 Pages 10 B/W Illustrations
    by Chapman & Hall

    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.

    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
    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
    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

    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
    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

    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
    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


    Exercises appear at the end of each chapter.


    Sudipto Banerjee, Anindya Roy

    "… a unique and remarkable book … has much to offer that is not found elsewhere. … In Linear Algebra and Matrix Analysis for Statistics, Sudipto Bannerjee and Anindya Roy have raised the bar for textbooks in this genre. For me, this book will be an invaluable resource for my teaching and research. … an outstanding choice for research-oriented statisticians who want a comprehensive theoretical treatment of the subject that will take them well beyond the prerequisites for the study of linear models."
    Journal of the American Statistical Association, Vol. 110, 2015

    "The sixteen chapters cover the full range of topics … Topics are presented in a logical order and in a reasonable pace. The book is compactly written and the approach throughout is rigorous, yet well readable. … an excellent introduction to linear algebra."
    Zentralblatt MATH 1309

    "This would be a reasonable candidate for use in a standard linear algebra course, even at institutions with no statistics majors. … The proofs are very detailed and the authors bind the argument together with clear text that flows beautifully. … Some linear algebra courses put a greater emphasis on concrete applications or on using software to get computations done. Other texts treat linear algebra as a branch of abstract algebra and allow spaces over arbitrary fields. This book is a strong contender for the vast majority of linear algebra courses that fall between those two extremes."
    MAA Reviews, October 2014

    "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