Book:Sudipto Banerjee/Linear Algebra and Matrix Analysis for Statistics

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Sudipto Banerjee and Anindya Roy: Applied Linear Algebra and Matrix Analysis for Statistics

Published $\text {2015}$, Chapman and Hall/CRC Press

ISBN 978-1-42009-538-8


This book is part of the Texts in Statistical Science series.


Subject Matter


Contents

1 Matrices, Vectors and Their Operations
1.1 Basic definitions and notation
1.2 Matrix addition and scalar-matrix multiplication
1.3 Matrix multiplication
1.4 Partitioned matrices
1.5 The trace of a square matrix
1.6 Some special matrices
1.7 Exercises
2 Systems of linear equations
2.1 Introduction
2.2 Gaussian elimination
2.3 Gauss-Jordan elimination
2.4 Elementary matrices
2.5 Homogeneous linear systems
2.6 The inverse of a matrix
2.7 Exercises
3 More on linear equations
3.1 The LU decomposition
3.2 Crout's algorithm
3.3 LU decomposition with row interchanges
3.4 The LDU and Cholesky factorizations
3.5 Inverse of partitioned matrices
3.6 The LDU decomposition for partitioned matrices
3.7 The Sherman-Woodbury-Morrison formula
3.8 Exercises
4 Euclidean spaces
4.1 Introduction
4.2 Vector addition and scalar multiplication
4.3 Linear spaces and subspaces
4.4 Intersection and sum of subspaces
4.5 Linear combination and spans
4.6 Four fundamental subspaces
4.7 Linear independence
4.8 Basis and dimension
4.9 Change of basis and similar matrices
4.10 Exercises
5 The rank of a matrix
5.1 Rank and nullity of a matrix
5.2 Bases for the four fundamental subspaces
5.3 Rank and inverse
5.4 Rank factorization
5.5 The rank normal form
5.6 Rank of a partitioned matrix
5.7 Bases for the fundamental subspaces using rank normal form
5.8 Exercises
6 Complementary subspaces
6.1 Sum of subspaces
6.2 The dimension of the sum of subspaces
6.3 Direct sums and complements
6.4 Projectors
6.5 The column space-null space decomposition
6.6 Invariant subspaces and the Core-Nilpotent decomposition
6.7 Exercises
7 Orthogonality, orthogonal subspaces and projections
7.1 Inner-products, norms and orthogonality
7.2 Row rank = column rank: A proof using orthogonality
7.3 Orthogonal projections
7.4 Gram-Schmidt orthogonalization
7.5 Orthocomplementary subspaces
7.6 The Fundamental Theorem of Linear Algebra
7.7 Exercises
8 More on orthogonality
8.1 Orthogonal matrices
8.2 The QR decomposition
8.3 Orthogonal projection and projector
8.4 Orthogoonal projector: Alternative derivations
8.5 Sum of orthogonal projectors
8.6 Orthogonal triangularization
8.7 Orthogonal similarity reduction to Hessenberg forms
8.8 Orthogonal reduction to bidiagonal forms
8.9 Some further reading on statistical linear models
8.10 Exercises
9 Revisiting linear equations
9.1 Introduction
9.2 Null spaces and the general solution of linear systems
9.3 Rank and linear systems
9.4 Generalized inverse of a matrix
9.5 Generalized inverses and linear systems
9.6 The Moore-Penrose inverse
9.7 Exercises
10 Determinants
10.1 Introduction
10.2 Some basic properties of determinants
10.3 Determinant of products
10.4 Computing determinants
10.5 The determinant of the transpose of a matrix---revisited
10.6 Determinants of partitioned matrices
10.7 Cofactors and expansion theorems
10.8 The minor and rank of a matrix
10.9 The Cauchy-Binet formula
10.10 The Laplace expansion
10.11 Exercises
11 Eigenvalues and eigenvectors
11.1 The eigenvalue equation
11.2 Characteristic polynomial and its roots
11.3 Eigenspaces and multiplicities
11.4 Diagonalizable matrices
11.5 Similarity with triangular matrices
11.6 Matrix polynomials and the Cayley-Hamilton Theorem
11.7 Spectral decomposition of real symmetric matrices
11.8 Computation of eigenvalues
11.9 Exercises
12 Singular value and Jordan decompositions
12.1 Singular value decomposition
12.2 The SVD and the four fundamental subspaces
12.3 SVD and linear systems
12.4 SVD, data compression and principal components
12.5 Computing the SVD
12.6 The Jordan Canonical Form
12.7 Implications of the Jordan Canonical Form
12.8 Exercises
13 Quadratic forms
12.1 Introduction
12.2 Quadratic forms
12.3 Matrices in quadratic forms
12.4 Positive and nonnegative definite matrices
12.5 Congruence and Sylvester's Law of Inertia
12.6 Nonnegative definite matrices and minors
12.7 Some inequalities related to quadratic forms
12.8 Simultaneous diagonalization and the generalized eigenvalue problem
12.9 Exercises
14 Kronecker product and related operations
14.1 Bilinear interpolation and the Kronecker product
14.2 Basic properties of Kronecker products
14.3 Inverses, rank and nonsingularity of Kronecker products
14.4 Matrix factorizations for Kronecker products
14.5 Eigenvalues and determinant
14.6 The vec and commutator operators
14.7 Linear systems involving Kronecker products
14.8 Sylvester's equation and the Kronecker sum
14.9 The Hadamard product
14.10 Exercises
15 Linear iterative systems, norms and convergence
15.1 Linear iterative systems and convergence of matrix powers
15.2 Vector norms
15.3 Spectral radius and matrix convergence
15.4 Matrix norms and the Gerschgorin circles
15.5 The singular value decomposition--revisited
15.6 Web page ranking and Markov chains
15.7 Iterative algorithms for solving linear equations
15.8 Exercises
16 Abstract Linear Algebra
16.1 General vector spaces
16.2 General inner products
16.3 Linear transformations, adjoint and rank
16.4 The four fundamental subspaces--revisited
16.5 Inverses of linear transformations
16.6 Linear transformations and matrices
16.7 Change of bases, equivalence and similar matrices
16.8 Hilbert spaces
16.9 Exercises
References
Index
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