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线性代数与矩阵：第二教程（影印版）

样章
作者：
Helene Shapiro
定价：
135.00 元
版面字数：
571千字
开本：
16开
装帧形式：
精装
版次：
1
最新版次
印刷时间：
2022年
ISBN：
978-7-04-057031-1
物料号：
57031-00
出版时间：
2022-02-28
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
代数学
暂无
  • 目录
    • 前辅文
      • 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 Cn
                                        • 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 Gerˇsgorin 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
                                                                                            • 6.2. Properties of Hermitian Matrices and Inertia
                                                                                              • 6.3. The Rayleigh–Ritz Ratio and the Courant–Fischer Theorem
                                                                                                • 6.4. Cauchy’s Interlacing Theorem and Other Eigenvalue Inequalities
                                                                                                  • 6.5. Positive Definite Matrices
                                                                                                    • 6.6. Simultaneous Row and Column Operations
                                                                                                      • 6.7. Hadamard’s Determinant Inequality
                                                                                                        • 6.8. Polar Factorization and Singular Value Decomposition
                                                                                                          • Exercises
                                                                                                          • Chapter 7. Vector and Matrix Norms
                                                                                                            • 7.1. Vector Norms
                                                                                                              • 7.2. Matrix Norms
                                                                                                                • Exercises
                                                                                                                • Chapter 8. Some Matrix Factorizations
                                                                                                                  • 8.1. Singular Value Decomposition
                                                                                                                    • 8.2. Householder Transformations
                                                                                                                      • 8.3. Using Householder Transformations to Get Triangular, Hessenberg, and Tridiagonal Forms
                                                                                                                        • 8.4. Some Methods for Computing Eigenvalues
                                                                                                                          • 8.5. LDU Factorization
                                                                                                                            • Exercises
                                                                                                                            • Chapter 9. Field of Values
                                                                                                                              • 9.1. Basic Properties of the Field of Values
                                                                                                                                • 9.2. The Field of Values for Two-by-Two Matrices
                                                                                                                                  • 9.3. Convexity of the Numerical Range
                                                                                                                                    • Exercises
                                                                                                                                    • Chapter 10. Simultaneous Triangularization
                                                                                                                                      • 10.1. Invariant Subspaces and Block Triangularization
                                                                                                                                        • 10.2. Simultaneous Triangularization, Property P, and Commutativity
                                                                                                                                          • 10.3. Algebras, Ideals, and Nilpotent Ideals
                                                                                                                                            • 10.4. McCoy’s Theorem
                                                                                                                                              • 10.5. Property L
                                                                                                                                                • Exercises
                                                                                                                                                • Chapter 11. Circulant and Block Cycle Matrices
                                                                                                                                                  • 11.1. The J Matrix
                                                                                                                                                    • 11.2. Circulant Matrices
                                                                                                                                                      • 11.3. Block Cycle Matrices
                                                                                                                                                        • Exercises
                                                                                                                                                        • Chapter 12. Matrices of Zeros and Ones
                                                                                                                                                          • 12.1. Introduction: Adjacency Matrices and Incidence Matrices
                                                                                                                                                            • 12.2. Basic Facts about (0, 1)-Matrices
                                                                                                                                                              • 12.3. The Minimax Theorem of K¨onig and Egerv´ary
                                                                                                                                                                • 12.4. SDRs, a Theorem of P. Hall, and Permanents
                                                                                                                                                                  • 12.5. Doubly Stochastic Matrices and Birkhoff’s Theorem
                                                                                                                                                                    • 12.6. A Theorem of Ryser
                                                                                                                                                                      • Exercises
                                                                                                                                                                      • Chapter 13. Block Designs
                                                                                                                                                                        • 13.1. t-Designs
                                                                                                                                                                          • 13.2. Incidence Matrices for 2-Designs
                                                                                                                                                                            • 13.3. Finite Projective Planes
                                                                                                                                                                              • 13.4. Quadratic Forms and the Witt Cancellation Theorem
                                                                                                                                                                                • 13.5. The Bruck–Ryser–Chowla Theorem
                                                                                                                                                                                  • Exercises
                                                                                                                                                                                  • Chapter 14. Hadamard Matrices
                                                                                                                                                                                    • 14.1. Introduction
                                                                                                                                                                                      • 14.2. The Quadratic Residue Matrix and Paley’s Theorem
                                                                                                                                                                                        • 14.3. Results of Williamson
                                                                                                                                                                                          • 14.4. Hadamard Matrices and Block Designs
                                                                                                                                                                                            • 14.5. A Determinant Inequality, Revisited
                                                                                                                                                                                              • Exercises
                                                                                                                                                                                              • Chapter 15. Graphs
                                                                                                                                                                                                • 15.1. Definitions
                                                                                                                                                                                                  • 15.2. Graphs and Matrices
                                                                                                                                                                                                    • 15.3. Walks and Cycles
                                                                                                                                                                                                      • 15.4. Graphs and Eigenvalues
                                                                                                                                                                                                        • 15.5. Strongly Regular Graphs
                                                                                                                                                                                                          • Exercises
                                                                                                                                                                                                          • Chapter 16. Directed Graphs
                                                                                                                                                                                                            • 16.1. Definitions
                                                                                                                                                                                                              • 16.2. Irreducibility and Strong Connectivity
                                                                                                                                                                                                                • 16.3. Index of Imprimitivity
                                                                                                                                                                                                                  • 16.4. Primitive Graphs
                                                                                                                                                                                                                    • Exercises
                                                                                                                                                                                                                    • Chapter 17. Nonnegative Matrices
                                                                                                                                                                                                                      • 17.1. Introduction
                                                                                                                                                                                                                        • 17.2. Preliminaries
                                                                                                                                                                                                                          • 17.3. Proof of Perron’s Theorem
                                                                                                                                                                                                                            • 17.4. Nonnegative Matrices
                                                                                                                                                                                                                              • 17.5. Irreducible Matrices
                                                                                                                                                                                                                                • 17.6. Primitive and Imprimitive Matrices
                                                                                                                                                                                                                                  • Exercises
                                                                                                                                                                                                                                  • Chapter 18. Error-Correcting Codes
                                                                                                                                                                                                                                    • 18.1. Introduction
                                                                                                                                                                                                                                      • 18.2. The Hamming Code
                                                                                                                                                                                                                                        • 18.3. Linear Codes: Parity Check and Generator Matrices
                                                                                                                                                                                                                                          • 18.4. The Hamming Distance
                                                                                                                                                                                                                                            • 18.5. Perfect Codes and the Generalized Hamming Code
                                                                                                                                                                                                                                              • 18.6. Decoding
                                                                                                                                                                                                                                                • 18.7. Codes and Designs
                                                                                                                                                                                                                                                  • 18.8. Hadamard Codes
                                                                                                                                                                                                                                                    • Exercises
                                                                                                                                                                                                                                                    • Chapter 19. Linear Dynamical Systems
                                                                                                                                                                                                                                                      • 19.1. Introduction
                                                                                                                                                                                                                                                        • 19.2. A Population Cohort Model
                                                                                                                                                                                                                                                          • 19.3. First-Order, Constant Coefficient, Linear Differential and Difference Equations
                                                                                                                                                                                                                                                            • 19.4. Constant Coefficient, Homogeneous Systems
                                                                                                                                                                                                                                                              • 19.5. Constant Coefficient, Nonhomogeneous Systems; Equilibrium Points
                                                                                                                                                                                                                                                                • 19.6. Nonnegative Systems
                                                                                                                                                                                                                                                                  • 19.7. Markov Chains
                                                                                                                                                                                                                                                                  • Exercises
                                                                                                                                                                                                                                                                    • Bibliography
                                                                                                                                                                                                                                                                      • Index

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