Linear algebra and matrix theory are fundamental tools in mathematical and physical science, as well as fertile fields for research. This second edition of this acclaimed text presents results of both classic and recent matrix analysis using canonical forms as a unifying theme and demonstrates their importance in a variety of applications. This thoroughly revised and updated second edition is a text for a second course on linear algebra and has more than 1,100 problems and exercises, new sections on the singular value and CS decompositions and the Weyr canonical form, expanded treatments of inverse problems and of block matrices, and much more.
内容简介:

本书从数学分析的角度阐述了矩阵分析的经典和现代方法，不仅包括由于数学分析的需要而产生的线性代数的论题，还广泛选择了其他相关学科如微分方程、最优化、逼近理论、工程学和运筹学等有关的论题。本书主要内容有：特征值、特征向量和相似性、酉相似、Schur三角化及其推论、正规矩阵、标准形和包括Jordan标准形在内的各种分解、LU分解、QR分解和酉矩阵、Hermite矩阵和复对称矩阵、向量范数和矩阵范数、特征值的估计和扰动、正定矩阵、非负矩阵。

本书由美国著名数学家R.A.Horn教授和C．R．Johnson教授合著，是矩阵理论方面的经典著作，原书自1985年出版以来，已经重印了10余次。书中论述了矩阵分析的经典方法和现代方法，不仅涵盖了几乎所有的基础理论，还广泛地对涉及其他相关学科的各种论题进行了有效的阐述，并对有关论题提供了现代的参考资料。

本书逻辑清晰，结构严谨，既注重教学又注重应用。在每一章的开始，作者都介绍几个应用来引入本章的论题以激发学习兴趣。在章节末尾，作者还独具匠心地编排了许多具有探索性和启发性的习题，引导读者提高描述和解决数学问题的能力。所以不论是对从事线性代数纯理论研究还是从事应用研究的人员，本书都是一本必备的参考书。
本书可作为理工科专业研究生或数学专业高年级本科生教材，也可供数学工作者和科技人员参考。
Contents
Chapter 0 Review and miscellanea 1
0.0 Introduction 1
0.1 Vector spaces 1
0.2 Matrices 4
0.3 Determinants 7
0.4 Rank 12
0.5 Nonsingularity 14
0.6 The usual inner product 14
0.7 Partitioned matrices 17
0.8 Determinants again 19
0.9 Special types of matrices 23
0.10 Change of basis 30
Chapter 1 Eigenvalues, eigenvectors, and similarity 33
1.0 Introduction 33
1.1 The eigenvalue-eigenvector equation 34
1.2 The characteristic polynomial 38
1.3 Similarity 44
1.4 Eigenvectors 57
Chapter 2 Unitary equivalence and normal matrices 65
2.0 Introduction 65
2.1 Unitary matrices 66
2.2 Unitary equivalence 72
2.3 Schur's unitary triangularizat ervations and applications 129
3.3 Polynomials and matrices: the minimal polynomial 142
3.4 Other canonical forms and factorizations 150
3.5 Triangular factorizations 158
Chapter 4 Hermitian and symmetric matrices 167
4.0 Introduction 167
4.1 Definitions, properties, and characterizations of Hermitian matrices 169
4.2 Variational characterizations of eigenvalues of Hermitian matrices 176
4.3 Some applications of the variational characterizations 181
4.4 Complex symmetric matrices 201
4.5 Congruence and simultaneous diagonalization of Hermitian and symmetric matrices 218
4.6 Consimilarity and condiagonalization 244
Chapter 5 Norms for vectors and matrices 257
5.0 Introduction 257
5.1 Defining properties of vector norms and inner products 259
5.2 Examples of vector norms 264
5.3 Algebraic properties of vector norms 268
5.4 Analytic properties of vector norms 269
5.5 Geometric properties of vector norms 281
5.6 Matrix norms 290
5.7 Vector norms on matrices 320
5.8 Errors in inverses and solutions of linear systems 335
Chapter 6 Location and perturbation of eigenvalues 343
6.0 Introduction 343
6.1 Gergorin discs 344
6.2 Gergorin discs - a closer look 353
6.3 Perturbation theorems 364
6.4 Other inclusion regions 378
Chapter 7 Positive definite matrices 391
7.0 Introduction 391
7.1 Definitions and properties 396
7.2 Characterizations 402
7.3 The polar form and the singular value decomposition 411
7.4 Examples and applications of the singular value decomposition 427
7.5 The Schur product theorem 455
7.6 Congruence: products and simultaneous diagonalization 464
7.7 The positive semidefinite ordering 469
7.8 Inequalities for positive definite matrices 476
Chapter 8 Nonnegative matrices 487
8.0 Introduction 487
8.1 Nonnegative matrices - inequalities and generalities 490
8.2 Positive matrices 495
8.3 Nonnegative matrices 503
8.4 Irreducible nonnegative matrices 507
8.5 Primitive matrices 515
8.6 A general limit theorem 524
8.7 Stochastic and doubly stochastic matrices 526
Appendices
A Complex numbers 531
B Convex sets and functions 533
C The fundamental theorem of algebra 537
D Continuous dependence of the zeroes of a olynomial on its coefficients 539
E Weierstrass's theorem 541
References 543
Notation 547
Index 549