应用矩阵分析导论（英文版）
作者： 金小庆、黄锡荣
出版时间：2016年4月
出版社：高等教育出版社
- 高等教育出版社
- 9787040449945
- 1版
- 46946
- 0044175625-1
- 16开
- 2016年4月
- 180
- 130
- 理学
- 数学
- O151.21
- 数学类
- 研究生、本科
本书包括8章，它包括扰动和误差分析：求解线性系统的共轭梯度法和预处理技术；基于正交变换的最小二乘法等。
最后的2章包括了该领域的一些最新进展。第7章构造了矩阵函数最优的预处理器。更确切地说，令f为一个矩阵函数。给定一个矩阵A，有两种选择构造，f（A）最佳预处理器。我们研究了不同矩阵函数预处理器的性质。第8章研究Bottcher-Wenzel猜想并讨论相关问题。
本书要求基础知识为各个学科都开设的基本的线性代数、微积分、数值分析和计算知识。本书可作为科学和工程系高年级本科生或者低年级研究生的教材。本书也可供对应用矩阵理论感兴趣的计算科学研究人员参考。
Preface
1. Introduction and Review
1.1 Basic symbols
1.2 Quadratic forms and positive definite matrices
1.2.1 Quadratic forms
1.2.2 Problems involving quadratic forms
1.2.3 Positive definite matrix
1.2.4 Other methods to determine the positive definiteness
1.3 Theorems for eigenvalues of symmetric matrices
1.4 Complex inner product spaces
1.5 Hermitian, unitary, and normal matrices
1.6 Kronecker product and Kronecker sum
2. Norms and Perturbation Analysis
2.1 Vector norms
2.2 Matrix norms
2.3 Perturbation analysis for linear systems
2.4 Error on floating point numbers
3. Least Squares Problems
3.1 Solution of LS problems
3.2 Perturbation analysis for LS problems
3.3 Orthogonal transformations
3.3.1 Householder reflections
3.3.2 Givens rotations
3.4 An algorithm based on QR factorization
3.4.1 QR factorization
3.4.2 A practical algorithm for LS problems
4. Generalized Inverses
4.1 Moore-Penrose generalized inverse
4.2 Basic properties
4.3 Relation to LS problems
4.4 Other generalized inverses
5. Conjugate Gradient Method
5.1 Steepest descent method
5.1.I Steepest descent method
5.1.2 Convergence rate
5.2 Conjugate gradient method
5.2.1 Conjugate gradient method
5.2.2 Basic properties
5.2.3 Practical CG method
5.3 Preconditioning technique
6. Optimal and Superoptimal Preconditioners
6.1 Introduction to optimal preconditioner
6.1.1 Circulant matrix
6.1.2 Optimal preconditioner
6.2 Linear operator cu
6.2.1 Algebraic properties
6.2.2 Geometric properties
6.3 Stability
6.4 Superoptimal preconditioner
6.5 Spectral relation of preconditioned matrices
7. Optimal Preconditioners for Functions of Matrices
7.1 Optimal preconditioners for matrix exponential
7.2 Optimal preconditioners for matrix cosine and matrix sine
7.3 Optimal preconditioners for matrix logarithm
8. BSttcher-Wenzel Conjecture and Related Problems
8.1 Introduction to BSttcher-Wenzel conjecture
8.2 The proof of B5ttcher-Wenzel conjecture
8.3 Maximal pairs of the inequality
8.4 Other related problems
8.4.1 The use of other norms in the inequality
8.4.2 The sharpening of the inequality
8.4.3 The extension to other products similar to the commutator
Bibliography
Index
