Linear algebra in action /
Harry Dym.
- Second edition.
- xix, 585 pages : illustrations ; 26 cm.
- Graduate studies in mathematics ; volume 78. .
- Graduate studies in mathematics ; volume 78. .
Includes bibliographical references (pages 575-578) and indexes.
Preview -- The abstract definition of a vector space -- Some definitions -- Mappings -- Triangular matrices -- Block triangular matrices -- Schur complements -- Other matrix products -- Some preliminary observations -- Examples -- Upper echelon matrices -- The conservation of dimension -- Quotient spaces -- Conservation of dimension for matrices -- From U to A -- Square matrices -- Gaussian elimination redux -- Properties of BA and AC -- Extracting a basis -- Computing the coefficients in a basis -- The Gauss-Seidel method -- Block Gaussian elimination -- -- Review -- Change of basis and similarity -- Invariant subspaces -- Existence of eigenvalues -- Eigenvalues for matrices -- Direct sums -- Diagonalizable matrices -- An algorithm for diagonalizing matrices. Machine generated contents note: 1.1. 1.2. 1.3. 1.4. 1.5. 1.6. 1.7. 1.8. 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8. 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.7. 3.8. 4.1. 4.2. 4.3. 4.4. 4.5. 4.6. 4.7. Computing eigenvalues at this point -- Not all matrices are diagonalizable -- The Jordan decomposition theorem -- An instructive example -- The binomial formula -- More direct sum decompositions -- Verification of Theorem 4.13 -- Bibliographical notes -- Functionals -- Determinants -- Useful rules for calculating determinants -- Eigenvalues -- Exploiting block structure -- The Binet-Cauchy formula -- Minors -- Uses of determinants -- Companion matrices -- Circulants and Vandermonde matrices -- Overview -- Structure of the nullspaces NBj -- Chains and cells -- Computing J -- An algorithm for computing U -- A simple example -- A more elaborate example -- Jordan decompositions for real matrices -- Projection matrices -- Companion and generalized Vandermonde matrices -- Four inequalities -- Normed linear spaces -- Equivalence of norms -- Norms of linear transformations. 4.8. 4.9. 4.10. 4.11. 4.12. 4.13. 4.14. 4.15. 5.1. 5.2. 5.3. 5.4. 5.5. 5.6. 5.7. 5.8. 5.9. 5.10. 6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7. 6.8. 6.9. 6.10. 7.1. 7.2. 7.3. 7.4. Operator norms for matrices -- Mixing tops and bottoms -- Evaluating some operator norms -- Inequalities for multiplicative norms -- Small perturbations -- Bounded linear functionals -- Extensions of bounded linear functionals -- Banach spaces -- Bibliographical notes -- Inner product spaces -- A characterization of inner product spaces -- Orthogonality -- Gram matrices -- Projections and direct sum decompositions -- Orthogonal projections -- Orthogonal expansions -- The Gram-Schmidt method -- Toeplitz and Hankel matrices -- Adjoints -- The Riesz representation theorem -- Normal, selfadjoint and unitary transformations -- Auxiliary formulas -- Gaussian quadrature -- Bibliographical notes -- Hermitian matrices are diagonalizable -- Commuting Hermitian matrices -- Real Hermitian matrices -- Projections and direct sums in Fn -- Projections and rank. 7.5. 7.6. 7.7. 7.8. 7.9. 7.10. 7.11. 7.12. 7.13. 8.1. 8.2. 8.3. 8.4. 8.5. 8.6. 8.7. 8.8. 8.9. 8.10. 8.11. 8.12. 8.13. 8.14. 8.15. 9.1. 9.2. 9.3. 9.4. 9.5. Normal matrices -- QR factorization -- Schur's theorem -- Areas, volumes and determinants -- Boundary value problems -- Bibliographical notes -- Singular value decompositions -- Complex symmetric matrices -- Approximate solutions of linear equations -- Fitting a line in R2 -- Fitting a line in Rp -- Projection by iteration -- The Courant-Fischer theorem -- Inequalities for singular values -- von Neumann's inequality for contractive matrices -- Bibliographical notes -- Pseudoinverses -- The Moore-Penrose inverse -- Best approximation in terms of Moore-Penrose inverses -- Drazin inverses -- Bibliographical notes -- A detour on triangular factorization -- Definite and semidefinite matrices -- Characterizations of positive definite matrices -- An application of factorization -- Positive definite Toeplitz matrices -- Detour on block Toeplitz matrices. 9.6. 9.7. 9.8. 9.9. 9.10. 9.11. 10.1. 10.2. 10.3. 10.4. 10.5. 10.6. 10.7. 10.8. 10.9. 10.10. 11.1. 11.2. 11.3. 11.4. 11.5. 12.1. 12.2. 12.3. 12.4. 12.5. 12.6. A maximum entropy matrix completion problem -- A class of A 0 for which (12.52) holds -- Schur complements for semidefinite matrices -- Square roots -- Polar forms -- Matrix inequalities -- A minimal norm completion problem -- A description of all solutions to the minimal norm completion problem -- Bibliographical notes -- Systems of difference equations -- Nonhomogeneous systems of difference equations -- The exponential etA -- Systems of differential equations -- Uniqueness -- Isometric and isospectral flows -- Second-order differential systems -- Stability -- Nonhomogeneous differential systems -- Strategy for equations -- Second-order difference equations -- Higher order difference equations -- Second-order differential equations -- Higher order differential equations -- Wronskians -- Variation of parameters -- Mean value theorems. 12.7. 12.8. 12.9. 12.10. 12.11. 12.12. 12.13. 12.14. 12.15. 13.1. 13.2. 13.3. 13.4. 13.5. 13.6. 13.7. 13.8. 13.9. 13.10. 13.11. 13.12. 13.13. 13.14. 13.15. 13.16. 14.1. Taylor's formula with remainder -- Application of Taylor's formula with remainder -- Mean value theorem for functions of several variables -- Mean value theorems for vector-valued functions of several variables -- A contractive fixed point theorem -- Newton's method -- A refined contractive fixed point theorem -- Spectral radius -- The Brouwer fixed point theorem -- Bibliographical notes -- Preliminary discussion -- The implicit function theorem -- A generalization of the implicit function theorem -- Continuous dependence of solutions -- The inverse function theorem -- Roots of polynomials -- An instructive example -- A more sophisticated approach -- Dynamical systems -- Lyapunov functions -- Bibliographical notes -- Classical extremal problems -- Convex functions -- Extremal problems with constraints -- Examples -- Krylov subspaces. 14.2. 14.3. 14.4. 14.5. 14.6. 14.7. 14.8. 14.9. 14.10. 14.11. 15.1. 15.2. 15.3. 15.4. 15.5. 15.6. 15.7. 15.8. 15.9. 15.10. 15.11. 16.1. 16.2. 16.3. 16.4. 16.5.