科学和工程中的大部分问题最终将纳入矩阵问题。本书提供了应用矩阵理论基础介绍，也包括最近几年的一些新的结论。
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。
最后的二章包括了该领域的一些最新进展。在第7章，我们构造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的预处理器的性质。在第8章，我们研究Bottcher-Wenzel猜想并讨论相关问题。
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。 本书也可作为对应用矩阵理论感兴趣的计算科学研究人员参考。
- Preface vii
- 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
- 5.1.1 Steepest descent method
- 5.1.2 Convergence rate
- 5.2 Conjugate gradientmethod
- 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 Circulantmatrix
- 6.1.2 Optimal preconditioner
- 6.2 Linear operator c_U
- 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. Böttcher-Wenzel Conjecture and Related Problems
- 8.1 Introduction to Böttcher-Wenzel conjecture
- 8.2 The proof of Böttcher-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
