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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
ISBN：
978-7-04-057031-1
版面字数：
571.000千字
开本：
16开
全书页数：
暂无
装帧形式：
精装
重点项目：
暂无
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
  • 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
    • 19.6. Nonnegative Systems
    • 19.7. Markov Chains
  • Exercises
  • Bibliography
  • Index

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