Matrix algebra is one of the most important areas of mathematics for data analysis and for statistical theory. The first part of this book presents the relevant aspects of the theory of matrix algebra for applications in statistics. This part begins with the fundamental concepts of vectors and vector spaces, next covers the basic algebraic properties of matrices, then describes the analytic properties of vectors and matrices in the multivariate calculus, and finally discusses operations on matrices in solutions of linear systems and in eigenanalysis. This part is essentially self-contained.

The second part of the book begins with a consideration of various types of matrices encountered in statistics, such as projection matrices and positive definite matrices, and describes the special properties of those matrices. The second part also describes some of the many applications of matrix theory in statistics, including linear models, multivariate analysis, and stochastic processes. The brief coverage in this part illustrates the matrix theory developed in the first part of the book. The first two parts of the book can be used as the text for a course in matrix algebra for statistics students, or as a supplementary text for various courses in linear models or multivariate statistics.

The third part of this book covers numerical linear algebra. It begins with a discussion of the basics of numerical computations, and then describes accurate and efficient algorithms for factoring matrices, solving linear systems of equations, and extracting eigenvalues and eigenvectors. Although the book is not tied to any particular software system, it describes and gives examples of the use of modern computer software for numerical linear algebra. This part is essentially self-contained, although it assumes some ability to program in Fortran or C and/or the ability to use R/S-Plus or Matlab. This part of the book can be used as the text for a course in statistical computing, or as a supplementary text for various courses that emphasize computations.

The book includes a large number of exercises with some solutions provided in an appendix.

矩阵代数是数据分析和统计理论中最重要的数学领域之一。本书的第一部分介绍了矩阵代数理论在统计学中应用的相关内容。这一部分从向量和向量空间的基本概念开始，接下来涵盖了矩阵的基本代数性质，然后描述了多变量微积分中向量和矩阵的分析性质，最后讨论了在线性系统的解和特征分析中对矩阵的运算。这一部分基本上是自成一体的。

本书的第二部分首先考虑了统计学中遇到的各种类型的矩阵，如投影矩阵和正定矩阵，并描述了这些矩阵的特殊性质。第二部分还介绍了矩阵理论在统计学中的一些诸多应用，包括线性模型、多变量分析和随机过程。这一部分的简要介绍说明了本书第一部分中发展的矩阵理论。本书的前两部分可以作为统计学学生的矩阵代数课程的教材，也可以作为线性模型或多元统计学的各种课程的补充教材。

本书的第三部分涉及数值线性代数。它首先讨论了数值计算的基础知识，然后描述了准确和有效的算法，用于因子矩阵、解线性方程组以及提取特征值和特征向量。尽管本书没有与任何特定的软件系统挂钩，但它描述了现代计算机软件在数值线性代数中的使用并给出了实例。这一部分基本上是自成一体的，尽管它假定有一些用Fortran或C语言编程的能力和/或使用R/S-Plus或Matlab的能力。本书的这一部分可以作为统计计算课程的文本，或作为强调计算的各种课程的补充文本。

本书包括大量的练习，并在附录中提供了一些解决方案。