TY  - BOOK
AU  - Dym,H.
TI  - Linear algebra in action
T2  - Graduate studies in mathematics
SN  - 9781470409081 (alkaline paper)
PY  - 2013///]
CY  - Providence, Rhode Island
PB  - American Mathematical Society
KW  - Algebras, Linear
N1  - 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
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