统计学中的矩阵代数理论与应用详解（第二版）

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《矩阵代数理论、计算与统计应用》第二版是一本由James E. Gentle所著的专业书籍，它属于SpringerTexts in Statistics系列。该书全面探讨了矩阵代数的基础理论、计算方法及其在统计学领域的实际应用。作者以其丰富的经验和深入理解，将复杂的数学概念转化为易于理解的实例，旨在帮助读者掌握矩阵运算在统计分析中的核心作用。本书涵盖了多个关键主题，包括但不限于： 1. 矩阵理论：介绍了矩阵的基本概念，如向量空间、矩阵运算（加法、乘法、逆矩阵、转置等）、特征值和特征向量、秩和行列式等，这些都是后续计算和应用的基础。 2. 矩阵导数：对于那些涉及优化问题的统计模型，矩阵导数的计算至关重要。书中详细阐述了如何对矩阵函数求导，这对于梯度下降法、最大似然估计等常用算法的理解至关重要。 3. 统计应用：矩阵代数在统计学中有广泛的应用，比如多元统计分析中的主成分分析（PCA）、因子分析、协方差和相关矩阵的处理、线性回归模型的建模、以及更高级的统计模型，如线性混合效应模型中的矩阵形式。 4. 数值计算：书中还讨论了如何有效地进行矩阵运算和求解大规模线性系统，这对计算机实现矩阵操作具有指导意义，特别是在大数据背景下。 5. 最新版本更新：作为第二版，本书在第一版的基础上进行了修订和补充，反映了近年来矩阵理论和统计学领域的新进展，确保了内容的时效性和实用性。 6. 版权信息：该书享有严格的版权保护，所有权利归Springer International Publishing AG所有，禁止未经许可的复制、再版、传播或电子存储等行为。《Matrix Algebra Theory, Computations and Applications in Statistics》第二版是统计学专业学生和研究人员不可或缺的参考书，无论是在理论学习还是实际项目中，都能提供强大的工具和深入的洞察力。通过阅读这本书，读者将能够理解和掌握矩阵代数的核心原理，并将其应用于解决实际的统计问题。

Contents

Preface to the Second Edition ................................. vii

Preface to the First Edition ................................... ix

Part I Linear Algebra

1 Basic Vector/Matrix Structure and Notation .............. 3

1.1 Vectors ................................................. 4

1.2 Arrays ................................................. 5

1.3 Matrices ............................................... 5

1.3.1 Subvectors and Submatrices ........................ 8

1.4 Representation of Data ................................... 8

2 Vectors and Vector Spaces ................................. 11

2.1 Operations on Vectors ................................... 11

2.1.1 Linear Combinations and Linear Independence ........ 12

2.1.2 Vector Spaces and Spaces of Vectors ................. 13

2.1.3 Basis Sets for Vector Spaces ........................ 21

2.1.4 Inner Products .................................... 23

2.1.5 Norms ........................................... 25

2.1.6 Normalized Vectors ................................ 31

2.1.7 Metrics and Distances ............................. 32

2.1.8 Orthogonal Vectors and Orthogonal Vector Spaces ..... 33

2.1.9 The “One Vector” ................................. 34

2.2 Cartesian Coordinates and Geometrical Properties of Vectors .35

2.2.1 Cartesian Geometry ............................... 36

2.2.2 Projections ....................................... 36

2.2.3 Angles Between Vectors ............................ 37

2.2.4 Orthogonalization Transformations: Gram-Schmidt .... 38

2.2.5 Orthonormal Basis Sets ............................ 40

xvii

xviii Contents

2.2.6 Approximation of Vectors .......................... 41

2.2.7 Flats, Aﬃne Spaces, and Hyperplanes ................ 43

2.2.8 Cones ............................................ 43

2.2.9 Cross Products in IR

.............................. 46

2.3 Centered Vectors and Variances and Covariances of Vectors ... 48

2.3.1 The Mean and Centered Vectors .................... 48

2.3.2 The Standard Deviation, the Variance, and

Scaled Vectors .................................... 49

2.3.3 Covariances and Correlations Between Vectors ........ 50

Exercises ................................................... 52

3 Basic Properties of Matrices ............................... 55

3.1 Basic Deﬁnitions and Notation ............................ 55

3.1.1 Multiplication of a Matrix by a Scalar ............... 56

3.1.2 Diagonal Elements: diag(·) and vecdiag(·) ............ 56

3.1.3 Diagonal, Hollow, and Diagonally Dominant Matrices .. 57

3.1.4 Matrices with Special Patterns of Zeroes ............. 58

3.1.5 Matrix Shaping Operators .......................... 59

3.1.6 Partitioned Matrices ............................... 61

3.1.7 Matrix Addition................................... 63

3.1.8 Scalar-Valued Operators on Square Matrices:

The Trace ........................................ 65

3.1.9 Scalar-Valued Operators on Square Matrices:

The Determinant .................................. 66

3.2 Multiplication of Matrices and Multiplication of

Vectors and Matrices .................................... 75

3.2.1 Matrix Multiplication (Cayley) ...................... 75

3.2.2 Multiplication of Matrices with Special Patterns ....... 78

3.2.3 Elementary Operations on Matrices .................. 80

3.2.4 The Trace of a Cayley Product That Is Square ........ 88

3.2.5 The Determinant of a Cayley Product of Square

Matrices ......................................... 88

3.2.6 Multiplication of Matrices and Vectors ............... 89

3.2.7 Outer Products ................................... 90

3.2.8 Bilinear and Quadratic Forms: Deﬁniteness ........... 91

3.2.9 Anisometric Spaces ................................ 93

3.2.10 Other Kinds of Matrix Multiplication ................ 94

3.3 Matrix Rank and the Inverse of a Matrix ................... 99

3.3.1 Row Rank and Column Rank .......................100

3.3.2 Full Rank Matrices ................................101

3.3.3 Rank of Elementary Operator Matrices and Matrix

Products Involving Them...........................101

3.3.4 The Rank of Partitioned Matrices, Products of

Matrices, and Sums of Matrices .....................102

Contents xix

3.3.5 Full Rank Partitioning .............................104

3.3.6 Full Rank Matrices and Matrix Inverses ..............105

3.3.7 Full Rank Factorization ............................109

3.3.8 Equivalent Matrices ...............................110

3.3.9 Multiplication by Full Rank Matrices ................112

3.3.10 Gramian Matrices: Products of the Form A

A ........115

3.3.11 A Lower Bound on the Rank of a Matrix Product . . . . . 117

3.3.12 Determinants of Inverses ...........................117

3.3.13 Inverses of Products and Sums of Nonsingular

Matrices .........................................118

3.3.14 Inverses of Matrices with Special Forms ..............120

3.3.15 Determining the Rank of a Matrix ...................121

3.4 More on Partitioned Square Matrices:

The Schur Complement ..................................121

3.4.1 Inverses of Partitioned Matrices .....................122

3.4.2 Determinants of Partitioned Matrices ................122

3.5 Linear Systems of Equations ..............................123

3.5.1 Solutions of Linear Systems .........................123

3.5.2 Null Space: The Orthogonal Complement .............126

3.6 Generalized Inverses .....................................127

3.6.1 Immediate Properties of Generalized Inverses .........127

3.6.2 Special Generalized Inverses: The Moore-Penrose

Inverse ...........................................127

3.6.3 Generalized Inverses of Products and Sums

of Matrices .......................................130

3.6.4 Generalized Inverses of Partitioned Matrices ..........131

3.7 Orthogonality ...........................................131

3.7.1 Orthogonal Matrices: Deﬁnition and

Simple Properties .................................132

3.7.2 Orthogonal and Orthonormal Columns ...............133

3.7.3 The Orthogonal Group .............................133

3.7.4 Conjugacy ........................................134

3.8 Eigenanalysis: Canonical Factorizations ....................134

3.8.1 Eigenvalues and Eigenvectors Are Remarkable ........135

3.8.2 Left Eigenvectors ..................................135

3.8.3 Basic Properties of Eigenvalues and Eigenvectors ......136

3.8.4 The Characteristic Polynomial ......................138

3.8.5 The Spectrum ....................................141

3.8.6 Similarity Transformations .........................146

3.8.7 Schur Factorization ................................147

3.8.8 Similar Canonical Factorization: Diagonalizable

Matrices .........................................148

3.8.9 Properties of Diagonalizable Matrices ................152

3.8.10 Eigenanalysis of Symmetric Matrices .................153

xx Contents

3.8.11 Positive Deﬁnite and Nonnegative Deﬁnite Matrices . . . 159

3.8.12 Generalized Eigenvalues and Eigenvectors ............160

3.8.13 Singular Values and the Singular Value Decomposition

(SVD) ...........................................161

3.9 Matrix Norms ...........................................164

3.9.1 Matrix Norms Induced from Vector Norms ...........165

3.9.2 The Frobenius Norm—The “Usual” Norm ............167

3.9.3 Other Matrix Norms ...............................169

3.9.4 Matrix Norm Inequalities ...........................170

3.9.5 The Spectral Radius ...............................171

3.9.6 Convergence of a Matrix Power Series ................171

3.10 Approximation of Matrices ...............................175

3.10.1 Measures of the Diﬀerence Between Two Matrices . . . . . 175

3.10.2 Best Approximation with a Matrix of Given Rank . . . . . 176

Exercises ...................................................178

4 Vector/Matrix Derivatives and Integrals ...................185

4.1 Functions of Vectors and Matrices .........................186

4.2 Basics of Diﬀerentiation ..................................186

4.2.1 Continuity........................................188

4.2.2 Notation and Properties ............................188

4.2.3 Diﬀerentials ......................................190

4.3 Types of Diﬀerentiation ..................................190

4.3.1 Diﬀerentiation with Respect to a Scalar ..............190

4.3.2 Diﬀerentiation with Respect to a Vector ..............191

4.3.3 Diﬀerentiation with Respect to a Matrix .............196

4.4 Optimization of Scalar-Valued Functions ...................198

4.4.1 Stationary Points of Functions ......................200

4.4.2 Newton’s Method .................................200

4.4.3 Least Squares .....................................202

4.4.4 Maximum Likelihood ..............................206

4.4.5 Optimization of Functions with Constraints ...........208

4.4.6 Optimization Without Diﬀerentiation ................213

4.5 Integration and Expectation: Applications to Probability

Distributions ............................................214

4.5.1 Multidimensional Integrals and Integrals Involving

Vectors and Matrices ..............................215

4.5.2 Integration Combined with Other Operations .........216

4.5.3 Random Variables and Probability Distributions ......217

Exercises ...................................................222

5 Matrix Transformations and Factorizations ................227

5.1 Factorizations ...........................................227

5.2 Computational Methods: Direct and Iterative ...............228

Contents xxi

5.3 Linear Geometric Transformations .........................229

5.3.1 Invariance Properties of Linear Transformations . . . . . . . 229

5.3.2 Transformations by Orthogonal Matrices .............230

5.3.3 Rotations ........................................231

5.3.4 Reﬂections .......................................233

5.3.5 Translations: Homogeneous Coordinates ..............234

5.4 Householder Transformations (Reﬂections) ..................235

5.4.1 Zeroing All Elements But One in a Vector ............236

5.4.2 Computational Considerations ......................237

5.5 Givens Transformations (Rotations) .......................238

5.5.1 Zeroing One Element in a Vector ....................239

5.5.2 Givens Rotations That Preserve Symmetry ...........240

5.5.3 Givens Rotations to Transform to Other Values .......240

5.5.4 Fast Givens Rotations .............................241

5.6 Factorization of Matrices .................................241

5.7 LU and LDU Factorizations ..............................242

5.7.1 Properties: Existence ..............................243

5.7.2 Pivoting..........................................246

5.7.3 Use of Inner Products ..............................247

5.7.4 Properties: Uniqueness .............................247

5.7.5 Properties of the LDU Factorization of a Square

Matrix ...........................................248

5.8 QR Factorization ........................................248

5.8.1 Related Matrix Factorizations .......................249

5.8.2 Matrices of Full Column Rank ......................249

5.8.3 Relation to the Moore-Penrose Inverse for Matrices of

Full Column Rank .................................250

5.8.4 Nonfull Rank Matrices .............................251

5.8.5 Relation to the Moore-Penrose Inverse ...............251

5.8.6 Determining the Rank of a Matrix ...................252

5.8.7 Formation of the QR Factorization ..................252

5.8.8 Householder Reﬂections to Form the

QR Factorization ..................................252

5.8.9 Givens Rotations to Form the QR Factorization .......253

5.8.10 Gram-Schmidt Transformations to Form the

QR Factorization ..................................254

5.9 Factorizations of Nonnegative Deﬁnite Matrices .............254

5.9.1 Square Roots .....................................254

5.9.2 Cholesky Factorization .............................255

5.9.3 Factorizations of a Gramian Matrix ..................258

5.10 Approximate Matrix Factorization .........................259

5.10.1 Nonnegative Matrix Factorization ...................259

5.10.2 Incomplete Factorizations ..........................260

Exercises ...................................................261

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