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线性代数实战（第二版）（影印版）

样章
作者：
Harry Dym
定价：
269.00元
ISBN：
978-7-04-063250-7
版面字数：
980.00千字
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：
2025-02-14
物料号：
63250-00
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
  • Preface to the Second Edition
  • Preface to the First Edition
  • Chapter 1. Vector spaces
    • §1.1. Preview
    • §1.2. The abstract definition of a vector space
    • §1.3. Some definitions
    • §1.4. Mappings
    • §1.5. Triangular matrices
    • §1.6. Block triangular matrices
    • §1.7. Schur complements
    • §1.8. Other matrix products
  • Chapter 2. Gaussian elimination
    • §2.1. Some preliminary observations
    • §2.2. Examples
    • §2.3. Upper echelon matrices
    • §2.4. The conservation of dimension
    • §2.5. Quotient spaces
    • §2.6. Conservation of dimension for matrices
    • §2.7. From U to A
    • §2.8. Square matrices
  • Chapter 3. Additional applications of Gaussian elimination
    • §3.1. Gaussian elimination redux
    • §3.2. Properties of BA and AC
    • §3.3. Extracting a basis
    • §3.4. Computing the coefficients in a basis
    • §3.5. The Gauss-Seidel method
    • §3.6. Block Gaussian elimination
    • §3.7. {0, 1, ∞}
    • §3.8. Review
  • Chapter 4. Eigenvalues and eigenvectors
    • §4.1. Change of basis and similarity
    • §4.2. Invariant subspaces
    • §4.3. Existence of eigenvalues
    • §4.4. Eigenvalues for matrices
    • §4.5. Direct sums
    • §4.6. Diagonalizable matrices
    • §4.7. An algorithm for diagonalizing matrices
    • §4.8. Computing eigenvalues at this point
    • §4.9. Not all matrices are diagonalizable
    • §4.10. The Jordan decomposition theorem
    • §4.11. An instructive example
    • §4.12. The binomial formula
    • §4.13. More direct sum decompositions
    • §4.14. Verification of Theorem 4.13
    • §4.15. Bibliographical notes
  • Chapter 5. Determinants
    • §5.1. Functionals
    • §5.2. Determinants
    • §5.3. Useful rules for calculating determinants
    • §5.4. Eigenvalues
    • §5.5. Exploiting block structure
    • §5.6. The Binet-Cauchy formula
    • §5.7. Minors
    • §5.8. Uses of determinants
    • §5.9. Companion matrices
    • §5.10. Circulants and Vandermonde matrices
  • Chapter 6. Calculating Jordan forms
    • §6.1. Overview
    • §6.2. Structure of the nullspaces NBj
    • §6.3. Chains and cells
    • §6.4. Computing J
    • §6.5. An algorithm for computing U
    • §6.6. A simple example
    • §6.7. A more elaborate example
    • §6.8. Jordan decompositions for real matrices
    • §6.9. Projection matrices
    • §6.10. Companion and generalized Vandermonde matrices
  • Chapter 7. Normed linear spaces
    • §7.1. Four inequalities
    • §7.2. Normed linear spaces
    • §7.3. Equivalence of norms
    • §7.4. Norms of linear transformations
    • §7.5. Operator norms for matrices
    • §7.6. Mixing tops and bottoms
    • §7.7. Evaluating some operator norms
    • §7.8. Inequalities for multiplicative norms
    • §7.9. Small perturbations
    • §7.10. Bounded linear functionals
    • §7.11. Extensions of bounded linear functionals
    • §7.12. Banach spaces
    • §7.13. Bibliographical notes
  • Chapter 8. Inner product spaces and orthogonality
    • §8.1. Inner product spaces
    • §8.2. A characterization of inner product spaces
    • §8.3. Orthogonality
    • §8.4. Gram matrices
    • §8.5. Projections and direct sum decompositions
    • §8.6. Orthogonal projections
    • §8.7. Orthogonal expansions
    • §8.8. The Gram-Schmidt method
    • §8.9. Toeplitz and Hankel matrices
    • §8.10. Adjoints
    • §8.11. The Riesz representation theorem
    • §8.12. Normal, selfadjoint and unitary transformations
    • §8.13. Auxiliary formulas
    • §8.14. Gaussian quadrature
    • §8.15. Bibliographical notes
  • Chapter 9. Symmetric, Hermitian and normal matrices
    • §9.1. Hermitian matrices are diagonalizable
    • §9.2. Commuting Hermitian matrices
    • §9.3. Real Hermitian matrices
    • §9.4. Projections and direct sums in Fn
    • §9.5. Projections and rank
    • §9.6. Normal matrices
    • §9.7. QR factorization
    • §9.8. Schur’s theorem
    • §9.9. Areas, volumes and determinants
    • §9.10. Boundary value problems
    • §9.11. Bibliographical notes
  • Chapter 10. Singular values and related inequalities
    • §10.1. Singular value decompositions
    • §10.2. Complex symmetric matrices
    • §10.3. Approximate solutions of linear equations
    • §10.4. Fitting a line in R2
    • §10.5. Fitting a line in Rp
    • §10.6. Projection by iteration
    • §10.7. The Courant-Fischer theorem
    • §10.8. Inequalities for singular values
    • §10.9. von Neumann’s inequality for contractive matrices
    • §10.10. Bibliographical notes
  • Chapter 11. Pseudoinverses
    • §11.1. Pseudoinverses
    • §11.2. The Moore-Penrose inverse
    • §11.3. Best approximation in terms of Moore-Penrose inverses
    • §11.4. Drazin inverses
    • §11.5. Bibliographical notes
  • Chapter 12. Triangular factorization and positive definite matrices
    • §12.1. A detour on triangular factorization
    • §12.2. Definite and semidefinite matrices
    • §12.3. Characterizations of positive definite matrices
    • §12.4. An application of factorization
    • §12.5. Positive definite Toeplitz matrices
    • §12.6. Detour on block Toeplitz matrices
    • §12.7. A maximum entropy matrix completion problem
    • §12.8. A class of A ＞ O for which (12.52) holds
    • §12.9. Schur complements for semidefinite matrices
    • §12.10. Square roots
    • §12.11. Polar forms
    • §12.12. Matrix inequalities
    • §12.13. A minimal norm completion problem
    • §12.14. A description of all solutions to the minimal norm completion problem
    • §12.15. Bibliographical notes
  • Chapter 13. Difference equations and differential equations
    • §13.1. Systems of difference equations
    • §13.2. Nonhomogeneous systems of difference equations
    • §13.3. The exponential etA
    • §13.4. Systems of differential equations
    • §13.5. Uniqueness
    • §13.6. Isometric and isospectral flows
    • §13.7. Second-order differential systems
    • §13.8. Stability
    • §13.9. Nonhomogeneous differential systems
    • §13.10. Strategy for equations
    • §13.11. Second-order difference equations
    • §13.12. Higher order difference equations
    • §13.13. Second-order differential equations
    • §13.14. Higher order differential equations
    • §13.15. Wronskians
    • §13.16. Variation of parameters
  • Chapter 14. Vector-valued functions
    • §14.1. Mean value theorems
    • §14.2. Taylor’s formula with remainder
    • §14.3. Application of Taylor’s formula with remainder
    • §14.4. Mean value theorem for functions of several variables
    • §14.5. Mean value theorems for vector-valued functions of several variables
    • §14.6. A contractive fixed point theorem
    • §14.7. Newton’s method
    • §14.8. A refined contractive fixed point theorem
    • §14.9. Spectral radius
    • §14.10. The Brouwer fixed point theorem
    • §14.11. Bibliographical notes
  • Chapter 15. The implicit function theorem
    • §15.1. Preliminary discussion
    • §15.2. The implicit function theorem
    • §15.3. A generalization of the implicit function theorem
    • §15.4. Continuous dependence of solutions
    • §15.5. The inverse function theorem
    • §15.6. Roots of polynomials
    • §15.7. An instructive example
    • §15.8. A more sophisticated approach
    • §15.9. Dynamical systems
    • §15.10. Lyapunov functions
    • §15.11. Bibliographical notes
  • Chapter 16. Extremal problems
    • §16.1. Classical extremal problems
    • §16.2. Convex functions
    • §16.3. Extremal problems with constraints
    • §16.4. Examples
    • §16.5. Krylov subspaces
    • §16.6. The conjugate gradient method
    • §16.7. Dual extremal problems
    • §16.8. Linear programming
    • §16.9. Bibliographical notes
  • Chapter 17. Matrix-valued holomorphic functions
    • §17.1. Differentiation
    • §17.2. Contour integration
    • §17.3. Evaluating integrals by contour integration
    • §17.4. A short detour on Fourier analysis
    • §17.5. The Hilbert matrix
    • §17.6. Contour integrals of matrix-valued functions
    • §17.7. Continuous dependence of the eigenvalues
    • §17.8. More on small perturbations
    • §17.9. Spectral radius redux
    • §17.10. Fractional powers
    • §17.11. Bibliographical notes
  • Chapter 18. Matrix equations
    • §18.1. The equation X−AXB = O
    • §18.2. The Sylvester equation AX−XB = C
    • §18.3. AX = XB
    • §18.4. Special classes of solutions
    • §18.5. Riccati equations
    • §18.6. Two lemmas
    • §18.7. An LQR problem
    • §18.8. Bibliographical notes
  • Chapter 19. Realization theory
    • §19.1. Minimal realizations
    • §19.2. Stabilizable and detectable realizations
    • §19.3. Reproducing kernel Hilbert spaces
    • §19.4. de Branges spaces
    • §19.5. Rα invariance
    • §19.6. A left tangential Nevanlinna-Pick interpolation problem
    • §19.7. Factorization of Θ(λ)
    • §19.8. Bibliographical notes
  • Chapter 20. Eigenvalue location problems
    • §20.1. Interlacing
    • §20.2. Sylvester’s law of inertia
    • §20.3. Congruence
    • §20.4. Counting positive and negative eigenvalues
    • §20.5. Exploiting continuity
    • §20.6. Gerˇsgorin disks
    • §20.7. The spectral mapping principle
    • §20.8. Inertia theorems
    • §20.9. An eigenvalue assignment problem
    • §20.10. Bibliographical notes
  • Chapter 21. Zero location problems
    • §21.1. Bezoutians
    • §21.2. The Barnett identity
    • §21.3. The main theorem on Bezoutians
    • §21.4. Resultants
    • §21.5. Other directions
    • §21.6. Bezoutians for real polynomials
    • §21.7. Stable polynomials
    • §21.8. Kharitonov’s theorem
    • §21.9. Bibliographical notes
  • Chapter 22. Convexity
    • §22.1. Preliminaries
    • §22.2. Convex functions
    • §22.3. Convex sets in Rn
    • §22.4. Separation theorems in Rn
    • §22.5. Hyperplanes
    • §22.6. Support hyperplanes
    • §22.7. Convex hulls
    • §22.8. Extreme points
    • §22.9. Brouwer’s theorem for compact convex sets
    • §22.10. The Minkowski functional
    • §22.11. The numerical range
    • §22.12. Eigenvalues versus numerical range
    • §22.13. The Gauss-Lucas theorem
    • §22.14. The Heinz inequality
    • §22.15. Extreme points for polyhedra
    • §22.16. Bibliographical notes
  • Chapter 23. Matrices with nonnegative entries
    • §23.1. Perron-Frobenius theory
    • §23.2. Stochastic matrices
    • §23.3. Behind Google
    • §23.4. Doubly stochastic matrices
    • §23.5. An inequality of Ky Fan
    • §23.6. The Schur-Horn convexity theorem
    • §23.7. Bibliographical notes
  • Appendix A. Some facts from analysis
    • §A.1. Convergence of sequences of points
    • §A.2. Convergence of sequences of functions
    • §A.3. Convergence of sums
    • §A.4. Sups and infs
    • §A.5. Topology
    • §A.6. Compact sets
    • §A.7. Normed linear spaces
  • Appendix B. More complex variables
    • §B.1. Power series
    • §B.2. Isolated zeros
    • §B.3. The maximum modulus principle
    • §B.4. ln (1−λ) when |λ| < 1
    • §B.5. Rouch´e’s theorem
    • §B.6. Liouville’s theorem
    • §B.7. Laurent expansions
    • §B.8. Partial fraction expansions
  • Bibliography
  • Notation Index
  • Subject Index

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