线性代数实战（第二版）（影印版）

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目录
- 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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线性代数实战（第二版）（影印版）

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