矩阵理论作为一种基本的数学工具，在数学与其他科学技术领域都有广泛应用。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。主要内容有：特征值、特征向量和相似性；酉相似和酉等价；相似标准型和三角分解；Hermite矩阵、对称矩阵和酉相合；向量范数和矩阵范数；特征值的估计和扰动；正定矩阵和半正定矩阵；正矩阵和非负矩阵。第2版进行了全面的修订和更新，用新的小节介绍了奇异值、CS分解和Weyr范式等其他内容，并附有1100多个线性代数课程的问题和练习。
线性代数和矩阵理论是数学和自然科学的基本工具，同时也是科学研究的沃土。本书是矩阵理论方面的经典著作，从数学分析的角度阐述了矩阵分析的经典和现代方法。主要内容有：特征值、特征向量和相似性；酉相似和酉等价；相似标准型和三角分解；Hermite矩阵、对称矩阵和酉相合；向量范数和矩阵范数；特征值的估计和扰动；正定矩阵和半正定矩阵；正矩阵和非负矩阵。第2版对第1版进行了全面的修订、更新和扩展。这一版不仅对基础线性代数和矩阵理论做了全面的总结，而且还新增了奇异值、CS分解和Weyr标准型的相关内容，扩展了与逆矩阵和分块矩阵相关的内容，介绍了Jordan标准型的新应用。此外，还附有1100多个问题和练习，并且给出了一些提示，以帮助读者提高解决数学问题的能力。本书可以用作本科生或者研究生的教材，也可用作数学工作者和科技人员的参考书。
Preface to the Second Edition page
Preface to the First Edition
0 Review and Miscellanea
0.0 Introduction
0.1 Vector spaces
0.2 Matrices
0.3 Determinants
0.4 Rank
0.5 Nonsingularity
0.6 The Euclidean inner product and norm
0.7 Partitioned sets and matrices
0.8 Determinants again
0.9 Special types of matrices
0.10 Change of basis
0.11 Equivalence relations
1 Eigenvalues, Eigenvectors, and Similarity
1.0 Introduction
1.1 The eigenvalue–eigenvector equation
1.2 The characteristic polynomial and algebraic multiplicity
1.3 Similarity
1.4 Left and right eigenvectors and geometric multiplicity
2 Unitary Similarity and Unitary Equivalence
2.0 Introduction
2.1 Unitary matrices and the QR factorization
2.2 Unitary similarity
2.3 Unitary and real orthogonal triangularizations
2.4 Consequences of Schur’s triangularization theorem
2.5 Normal matrices
2.6 Unitary equivalence and the singular value decomposition
2.7 The CS decomposition
3 Canonical Forms for Similarity and Triangular Factorizations
3.0 Introduction
3.1 The Jordan canonical form theorem
3.2 Consequences of the Jordan canonical form
3.3 The minimal polynomial and the companion matrix
3.4 The real Jordan and Weyr canonical forms
3.5 Triangular factorizations and canonical forms
4 Hermitian Matrices, Symmetric Matrices, and Congruences
4.0 Introduction
4.1 Properties and characterizations of Hermitian matrices
4.2 Variational characterizations and subspace intersections
4.3 Eigenvalue inequalities for Hermitian matrices
4.4 Unitary congruence and complex symmetric matrices
4.5 Congruences and diagonalizations
4.6 Consimilarity and condiagonalization
5 Norms for Vectors and Matrices
5.0 Introduction
5.1 Definitions of norms and inner products
5.2 Examples of norms and inner products
5.3 Algebraic properties of norms
5.4 Analytic properties of norms
5.5 Duality and geometric properties of norms
5.6 Matrix norms
5.7 Vector norms on matrices
5.8 Condition numbers: inverses and linear systems
6 Location and Perturbation of Eigenvalues
6.0 Introduction
6.1 Gergorin discs
6.2 Gergorin discs – a closer look
6.3 Eigenvalue perturbation theorems
6.4 Other eigenvalue inclusion sets
7 Positive Definite and Semidefinite Matrices
7.0 Introduction
7.1 Definitions and properties
7.2 Characterizations and properties
7.3 The polar and singular value decompositions
7.4 Consequences of the polar and singular value decompositions
7.5 The Schur product theorem
7.6 Simultaneous diagonalizations, products, and convexity
7.7 The Loewner partial order and block matrices
7.8 Inequalities involving positive definite matrices
8 Positive and Nonnegative Matrices
8.0 Introduction
8.1 Inequalities and generalities
8.2 Positive matrices
8.3 Nonnegative matrices
8.4 Irreducible nonnegative matrices
8.5 Primitive matrices
8.6 A general limit theorem
8.7 Stochastic and doubly stochastic matrices
Appendix A Complex Numbers
Appendix B Convex Sets and Functions
Appendix C The Fundamental Theorem of Algebra
Appendix D Continuity of Polynomial Zeroes and Matrix Eigenvalues
Appendix E Continuity, Compactness, andWeierstrass’s Theorem
Appendix F Canonical Pairs
References
Notation
Hints for Problems
Index
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