矩陣代數

矩陣代數是用於數據分析和統計領域的重要數學理論。本書《矩陣代數》是一部教科書，第一部分提供了在統計學中應用的矩陣代數理論，第2部分主要講應用，第3部分論及數值線性代數。本書各章有大量習題，書後附錄有題解或提示。可作為統計專業的矩陣代數教材。讀者對象：數理統計專業的學生教材。

Preface

Part Ⅰ Linear Algebra

1 Basic Vector/Matrix Structure and Notation

1.1 Vectors

1.2 Arrays

1.3 Matrices

1.4 Representation of Data

2 Vectors and Vector Spaces

2.1 Operations on Vectors

2.1.1 Linear Combinations and Linear Independence

2.1.2 Vector Spaces and Spaces of Vectors

2.1.3 Basis Sets

2.1.4 Inner Products

2.1.5 Norms

2.1.6 Normalized Vectors

2.1.7 Metrics and Distances

2.1.8 Orthogonal Vectors and Orthogonal Vector Spaces

2.1.9 The "One Vector"

2.2 Cartesian Coordinates and Geometrical Properties of Vectors

2.2.1 Cartesian Geometry

2.2.2 Projections

2.2.3 Angles between Vectors

2.2.4 Orthogonalization：Transformations

2.2.5 Orthonormal Basis Sets

2.2.6 Approximation of Vectors

2.2.7 Flats， Affine Spaces， and Hyperplanes

2.2.8 Cones

2.2.9 Cross Products in IR3

2.3 Centered Vectors and Variances and Covariances of Vectors

2.3.1 The Mean and Centered Vectors

2.3.2 The Standard Deviation， the Variance， and Scaled Vectors

2.3.3 Covariances and Correlations between Vectors

Exercises

3 Basic Properties of Matrices

3.1 Basic Definitions and Notation

3.1.1 Matrix Shaping Operators

3.1.2 Partitioned Matrices

3.1.3 Matrix Addition

3.1.4 Scalar—Valued Operators on Square Matrices： The Trace

3.1.5 Scalar—Valued Operators on Square Matrices： The Determinant

3.2 Multiplication of Matrices and Multiplication of Vectors and Matrices

3.2.1 Matrix Multiplication （Cayley）

3.2.2 Multiplication of Partitioned Matrices

3.2.3 Elementary Operations on Matrices

3.2.4 Traces and Determinants of Square Cayley Products

3.2.5 Multiplication of Matrices and Vectors

3.2.6 Outer Products

3.2.7 Bilinear and Quadratic Forms； Definiteness

3.2.8 Anisometric Spaces

3.2.9 Other Kinds of Matrix Multiplication

3.3 Matrix Rank and the Inverse of a Full Rank Matrix

3.3.1 The Rank of Partitioned Matrices， Products of Matrices， and Sums of Matrices

3.3.2 Full Rank Partitioning

3.3.3 Full Rank Matrices and Matrix Inverses

3.3.4 Full Rank Factorization

3.3.5 Equivalent Matrices

3.3.6 Multiplication by Full Rank Matrices

3.3.7 Products of the Form ATA

3.3.8 A Lower Bound on the Rank of a Matrix Product

3.3.9 Determinants of Inverses

3.3.10 Inverses of Products and Sums of Matrices

3.3.11 Inverses of Matrices with Special Forms

3.3.12 Determining the Rank of a Matrix

3.4 More on Partitioned Square Matrices： The Schur Complement

3.4.1 Inverses of Partitioned Matrices

3.4.2 Determinants of Partitioned Matrices

……

Part Ⅱ Applications in Data Analysis

Part Ⅲ Numerical Methods and Software

A Notation and Definitions

B Solutions and Hints for Selected Exercises

Bibliography

Index