线性代数渗透在数学的各个领域，或许比任何其他单一学科都更加广泛。它在纯数学与应用数学、统计学、计算机科学以及物理学和工程学的许多方面都发挥着至关重要的作用。本书以一种用户友好的方式传达了从实际分析师的角度看线性代数的基本与高级技术。通过广泛的应用和例子来展示这些技术，所选的例子突出展示了实用工具。本书适合本科生和研究生阅读。 In this new edition, most of the chapters in the first edition have been revised, some extensively. The revisions include changes in a number of proofs, either to simplify the argument, to make the logic clearer or, on occasion, to sharpen the result. New introductory sections on linear programming, extreme points for polyhedra and a Nevanlinna-Pick interpolation problem have been added, as have some very short introductory sections on the mathematics behind Google, Drazin inverses, band inverses and applications of SVD together with a number of new exercises.

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