Linear Algebra in Action

Transcription

1 2008 AGI-Information Management Consultants May be used for personal purporses only or by libraries associated to dandelon.com network. Linear Algebra in Action Harry Dym Graduate Studies in Mathematics Volume 78 American Mathematical Society Providence, Rhode Island

2 Contents Preface xv Chapter 1. Vector spaces Preview The abstract definition of a vector space Some definitions Mappings Triangular matrices Block triangular matrices Schur complements Other matrix products 19 Chapter 2. Gaussian elimination Some preliminary observations Examples Upper echelon matrices The conservation of dimension Quotient spaces Conservation of dimension for matrices From U to A Square matrices 41 Chapter 3. Additional applications of Gaussian elimination Gaussian elimination redux 45

3 vi Contents 3.2. Properties of BA and AC Extracting a basis Computing the coefficients in a basis The Gauss-Seidel method Block Gaussian elimination {0, 1, oo} Review 57 Chapter 4. Eigenvalues and eigenvectors Change of basis and similarity Invariant subspaces Existence of eigenvalues Eigenvalues for matrices Direct sums Diagonalizable matrices An algorithm for diagonalizing matrices 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 Bibliographical notes 87 Chapter 5. Determinants 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 109

4 Contents. vii Chapter 6. Calculating Jordan forms Overview Structure of the nullspaces J\f BJ Chains and cells Computing J An algorithm for U An example Another example Jordan decompositions for real matrices Companion and generalized Vandermonde matrices 128 Chapter 7. Normed linear spaces Four inequalities Normed linear spaces Equivalence of norms Norms of linear transformations Multiplicative norms Evaluating some operator norms Small perturbations Another estimate Bounded linear functional Extensions of bounded linear functionals Banach spaces 155 Chapter 8. Inner product spaces and orthogonality Inner product spaces A characterization of inner product spaces Orthogonality Gram matrices Adjoints The Riesz representation theorem Normal, selfadjoint and unitary transformations Projections and direct sum decompositions Orthogonal projections Orthogonal expansions The Gram-Schmidt method 177

5 viii Contents Toeplitz and Hankel matrices Gaussian quadrature Bibliographical notes 183 Chapter 9. Symmetric, Hermitian and normal matrices Hermitian matrices are diagonalizable Commuting Hermitian matrices Real Hermitian matrices Projections and direct sums in F n Projections and rank Normal matrices Schur's theorem QR factorization Areas, volumes and determinants Bibliographical notes 206 Chapter 10. Singular values and related inequalities Singular value decompositions Complex symmetric matrices Approximate solutions of linear equations The Courant-Fischer theorem Inequalities for singular values Bibliographical notes 225 Chapter 11. Pseudoinverses Pseudoinverses The Moore-Penrose inverse Best approximation in terms of Moore-Penrose inverses 237 Chapter 12. Triangular factorization and positive definite matrices A detour on triangular factorization Definite and semidefmite matrices Characterizations of positive definite matrices An application of factorization Positive definite Toeplitz matrices Detour on block Toeplitz matrices A maximum entropy matrix completion problem Schur complements for semidefinite matrices 262

6 Contents ix Square roots Polar forms Matrix inequalities A minimal norm completion problem A description of all solutions to the minimal norm completion problem Bibliographical notes 274 Chapter 13. Difference equations and differential equations Systems of difference equations The exponential e ta 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 Ordinary differential equations Wronskians Variation of parameters 295 Chapter 14. Vector valued functions Mean value theorems 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 Newton's method A contractive fixed point theorem A refined contractive fixed point theorem Spectral radius The Brouwer fixed point theorem Bibliographical notes 316

7 x. Contents Chapter 15. The implicit function theorem Preliminary discussion The main 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 336 Chapter 16. Extremal problems Classical extremal problems Extremal problems with constraints Examples Krylov subspaces The conjugate gradient method Dual extremal problems Bibliographical notes 356 Chapter 17. Matrix valued holomorphic functions Differentiation Contour integration Evaluating integrals by contour integration A short detour on Fourier analysis Contour integrals of matrix valued functions Continuous dependence of the eigenvalues More on small perturbations Spectral radius redux Fractional powers 381 Chapter 18. Matrix equations The equation X - AXB = C The Sylvester equation AX - XB = C Special classes of solutions 388

8 Contents. xi Chapter Chapter Riccati equations Two lemmas An LQR problem Bibliographical notes 19. Realization theory Minimal realizations Stabilizable and detectable realizations Reproducing kernel Hilbert spaces de Branges spaces R a invariance Factorization of (A) Bibliographical notes Chapter 20..Eigenvalue location problems Interlacing Sylvester's law of inertia Congruence Counting positive and negative eigenvalues Exploiting continuity Gersgorin disks The spectral mapping principle AX = XB Inertia theorems An eigenvalue assignment problem Bibliographical notes 21. Zero location problems Bezoutians A derivation of the formula for Hf based on realization The Barnett identity The main theorem on Bezoutians Resultants Other directions Bezoutians for real polynomials Stable polynomials Kharitonov's theorem

9 xii Contents Bibliographical notes 467 Chapter 22. Convexity Preliminaries Convex functions Convex sets in R n Separation theorems in R n Hyperplanes Support hyperplanes Convex hulls Extreme points Brouwer's theorem for compact convex sets The Minkowski functional The Gauss-Lucas theorem The numerical range Eigenvalues versus numerical range The Heinz inequality Bibliographical notes 494 Chapter 23. Matrices with nonnegative entries Perron-Frobenius theory Stochastic matrices Doubly stochastic matrices An inequality of Ky Fan The Schur-Horn convexity theorem Bibliographical notes 513 Appendix A. Some facts from analysis 515 A.l. Convergence of sequences of points 515 A.2. Convergence of sequences of functions 516 A.3. Convergence of sums 516 A.4. Sups and infs 517 A.5. Topology 518 A.6. Compact sets 518 A.7. Normed linear spaces 518 Appendix B. More complex variables 521 B.l. Power series 521

10 Contents - xiii B.2. Isolated zeros 523 B.3. The maximum modulus principle 525 B.4. In (1 - A) when A < B.5. Rouche's theorem 526 B.6. Liouville's theorem 528 B.7. Laurent expansions 528 B.8. Partial fraction expansions 529 Bibliography 531 Notation Index 535 Subject Index 537