Banerjee / Roy Linear Algebra and Matrix Analysis for Statistics

1. Auflage 2014
ISBN: 978-1-4822-4824-1
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 580 Seiten

Reihe: Chapman & Hall/CRC Texts in Statistical Science

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.

Zielgruppe

Graduate and advanced undergraduate students of statistics.

Autoren/Hrsg.

Banerjee, Sudipto

Roy, Anindya

Fachgebiete

• Mathematik | Informatik Mathematik Stochastik Mathematische Statistik
• Mathematik | Informatik Mathematik Algebra Lineare und multilineare Algebra, Matrizentheorie

Inhaltsverzeichnis

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.