Linear algebra in action / Harry Dym.

Includes bibliographical references (pages 575-578) and indexes.

Machine generated contents note: 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 -- 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 -- 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, infinity} -- 3.8. Review -- 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 -- 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 -- 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 -- 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 -- 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 -- 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 -- 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 -- 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 -- 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 0 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 -- 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 -- 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 -- 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 -- 16.1. Classical extremal problems -- 16.2. Convex functions -- 16.3. Extremal problems with constraints -- 16.4. Examples -- 16.5. Krylov subspaces.