Applied matrix algebra in the statistical sciences 统计学中的矩阵代数

This comprehensive text covers both applied and theoretical branches of matrix algebra in the statistical sciences. It also provides a bridge between linear algebra and statistical models. Appropriate for advanced undergraduate and graduate students, the self-contained treatment also constitutes a handy reference for researchers. The only mathematical background necessary is a sound knowledge of high school mathematics and a first course in statistics.
　　Consisting of two interrelated parts, this volume begins with the basic structure of vectors and vector spaces. The latter part emphasizes the diverse properties of matrices and their associated linear transformations and how these, in turn, depend upon results derived from linear vector spaces. An overview of introductory concepts leads to more advanced topics such as" latent roots and vectors, generalized inverses, and nonnegative matrices. Each chapter concludes with a section on real-world statistical applications, plus exercises that offer concrete examples of the applications of matrix algebra.

Preface
Chapter 1 Vectors
　1.1 Introduction
　1.2 Vector Operations
　　1.2.1 Vector Equality
　　1.2.2 Addition
　　1.2.3 Scalar Multiplication
　　1.2.4 Distributive Laws
　1.3 Coordinates of a Vector
　1.4 The Inner Product of Two Vectors
　　1.4.1 The Pythagorean Theorem
　　1.4.2 Length of a Vector and Distance Between Two Vectors
　　1.4.3 The Inner Product and Norm of a Vector
　1.5 The Dimension of a Vector: Unit Vectors
　1.6 Direction Cosines
　1.7 The Centroid of Vectors
　1.8 Metric and Normed Spaces
　　1.8.1 Minkowski Distance
　　1.8.2 Nonmetric Spaces
　1.9 Statistical Applications
　　1.9.1 The Mean Vector
　　1.9.2 Measures of Similarity
　Exercises
Chapter 2 Vector Spaces
　2.1 Introduction
　2.2 Vector Spaces
　2.3 The Dimension of a Vector Space
　2.4 The Sum and Direct Sum of a Vector Space
　2.5 0rthogonal Basis Vectors
　2.6 The Orthogonal Projection of a Vector
　　2.6.1 Gram-Schmidt Orthogonalization
　　2.6.2 The Area of a Parallelogram
　　2.6.3 Curve Fitting by Ordinary Least Squares
　2.7 Transformation of Coordinates
　　2.7.1 Orthogonal Rotation of Axes
　　2.7.2 Oblique Rotation of Axes
　　2.7.3 Curve Fitting by Rotating Axes
　Exercises
Chapter 3 Matrices and Systems of Linear Equations
　3.1 Introduction
　3.2 General Types of Matrices
　　3.2.1 Diagonal Matrix
　　3.2.2 Scalar and Unit Matrices
　　3.2.3 Incidence Matrix
　　3.2.4 Triangular Matrix
　　3.2.5 Symmetric and Transposed Matrices
　3.3 Matrix Operations
　　3.3.1 Addition
　　3.3.2 Multiplication
　　3.3.3 Matrix Transposition and Matrix Products
　　3.3.4 The K_ronecker and Hadamard Products
　3.4 Matrix Scalar Functions
　　3.4.1 The Permanent
　　3.4.2 The Determinant
　　3.4.3 The Determinant as Volume
　　3.4.4 The Trace
　　3.4.5 Matrix Rank
　3.5 Matrix Inversion
　　3.5.1 The Inverse of a Square Matrix
　　3.5.2 The Inverses of a Rectangular Matrix
　3.6 Elementary Matrices and Matrix Equivalence
　3.7 Linear Transformations and Systems of Linear Equations
　　3.7.1 Linear Transformations
　　3.7.2 Systems of Linear Equations
　Exercises
Chapter 4 Matrices of Special Type
　4.1 Symmetric Matrices
　4.2 Skew-Symmetric Matrices
　4.3 Positive Definite Matrices and Quadratic Forms
　4.4 Differentiation Involving Vectors and Matrices
　　4.4.1 Scalar Derivatives of Vectors
　　4.4.2 Vector Derivatives of Vectors
　　4.4.3 Extrema of Quadratic Forms
……
Chapter 5 Latent Roots and Latent Vectors
Chapter 6 Generalized Matrix Inverses
Chapter 7 Nonnegative and Diagonally
References
Index