线性代数与矩阵：第二教程（影印版）

线性代数和矩阵理论是 几乎每个数学领域（纯粹数 学和应用数学）的基本工具 。本书内容涵盖了核心主题 ，同时介绍了线性代数在其 中扮演关键角色的一些领域 ，例如区组设计、有向图、 纠错码和线性动力系统。本 书具有以下特色：讨论了 Weyr特征和Weyr典范形， 以及它们与更有名的， Jordan典范形的关系；利用 块循环矩阵和有向图来证明 非负不可约矩阵的特征值结 构上的Frobenius定理；包 含平衡不完全区组设计 （BIBDs）、Hadamard矩 阵和强正则图等组合论题。 此外本书还介绍了P-矩阵的 McCoy定理、关于区组设计 存在性的Bruck-Ryser- Chowla定理以及马尔可夫 链。本书是为熟悉线性代数 第一教程知识、有兴趣学习 更高级内容的读者编写的。

Preface Note to the Reader Chapter 1.Preliminaries 1.1.Vector Spaces 1.2.Bases and Coordinates 1.3.Linear Transformations 1.4.Matrices 1.5.The Matrix of a Linear Transformation 1.6.Change of Basis and Similarity 1.7.Transposes 1.8.Special Types of Matrices 1.9.Submatrices, Partitioned Matrices, and Block Multiplication 1.10.Invariant Subspaces 1.11.Determinants 1.12.Tensor Products Exercises Chapter 2.Inner Product Spaces and Orthogonality 2.1.The Inner Product 2.2.Length, Orthogonality, and Projection onto a Line 2.3.Inner Products in C” 2.4.Orthogonal Complements and Projection onto a Subspace 2.5.Hilbert Spaces and Fourier Series 2.6.Unitary Tranformations 2.7.The Gram-Schmidt Process and QR Factorization 2.8.Linear Functionals and the Dual Space Exercises Chapter 3.Eigenvalues, Eigenvectors, Diagonalization, and Triangularization 3.1.Eigenvalues 3.2.Algebraic and Geometric Multiplicity 3.3.Diagonalizability 3.4.A Triangularization Theorem 3.5.The Gersgorin Circle Theorem 3.6.More about the Characteristic Polynomial 3.7.Eigenvalues of AB and BA Exercises Chapter 4.The Jordan and Weyr Canonical Forms 4.1.A Theorem of Sylvester and Reduction to Block Diagonal Form 4.2.Nilpotent Matrices 4.3.The Jordan Form of a General Matrix 4.4.The Cayley-Hamilton Theorem and the Minimal Polynomial 4.5.Weyr Normal Form Exercises Chapter 5.Unitary Similarity and Normal Matrices 5.1.Unitary Similarity 5.2.Normal Matrices—the Spectral Theorem 5.3.More about Normal Matrices 5.4.Conditions for Unitary Similarity Exercises Chapter 6.Hermitian Matrices 6.1.Conjugate Bilinear Forms