FINITE-DIMENSIONAL LINEAR ALGEBRA

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1 DISCRETE MATHEMATICS AND ITS APPLICATIONS Series Editor KENNETH H ROSEN FINITE-DIMENSIONAL LINEAR ALGEBRA Mark S Gockenbach Michigan Technological University Houghton, USA CRC Press Taylor & Francis Croup Boca Raton London New York CRC Press is an Imprint of the Taylor & Francis Croup, an Informa business A CHAPMAN & HALL BOOK

2 Contents Preface xv About the author xxi 1 Some problems posed on vector spaces 1 11 Linear equations Systems of linear algebraic equations Linear ordinary differential equations Some interpretation: The structure of the solution set to a linear equation Finite fields and applications in discrete mathematics 7 12 Best approximation Overdetermined linear systems Best approximation by a polynomial Diagonalization Summary 17 2 Fields and vector spaces Fields Definition and examples Basic properties of fields Vector spaces Examples of vector spaces Subspaces Linear combinations and spanning sets Linear independence Basis and dimension Properties of bases Polynomial interpolation and the Lagrange basis Secret sharing Continuous piecewise polynomial functions Continuous piecewise linear functions Continuous piecewise quadratic functions Error in polynomial interpolation 90 ix

3 x Contents 3 Linear operators Linear operators Matrix operators More properties of linear operators Vector spaces of operators The matrix of a linear operator on Euclidean spaces Derivative and differential operators Representing spanning sets and bases using matrices The transpose of a matrix Isomorphic vector spaces Injective and surjective functions; inverses The matrix of a linear operator on general vector spaces Linear operator equations Homogeneous linear equations Inhomogeneous linear equations General solutions Existence and uniqueness of solutions The kernel of a linear operator and injectivity The rank of a linear operator and surjectivity Existence and uniqueness The fundamental theorem; inverse operators The inverse of a linear operator The inverse of a matrix Gaussian elimination Computing A' Fields other than R Newton's method Linear ordinary differential equations The dimension of ker(l) Finding a basis for ker(l) The easy case: Distinct real roots The case of repeated real roots The case of complex roots The Wronskian test for linear independence The Vandermonde matrix Graph theory The incidence matrix of a graph Walks and matrix multiplication Graph isomorphisms Coding theory Generator matrices; encoding and decoding Error correction The probability of errors Linear programming Specification of linear programming problems 184

4 Contents xi 3122 Basic theory The simplex method Finding an initial BPS Unbounded LPs Degeneracy and cycling Variations on the standard LPs Determinants and eigenvalues The determinant function Permutations The complete expansion of the determinant Further properties of the determinant function Practical computation of det(a) A recursive formula for det(a) Cramer's rule A note about polynomials Eigenvalues and the characteristic polynomial Eigenvalues of real matrix Diagonalization Eigenvalues of linear operators Systems of linear ODEs Complex eigenvalues Solving the initial value problem Linear systems in matrix form Integer programming Totally unimodular matrices Transportation problems The Jordan canonical form Invariant subspaces Direct sums Eigenspaces and generalized eigenspaces Generalized eigenspaces Appendix: Beyond generalized eigenspaces The Cayley-Hamilton theorem Nilpotent operators The Jordan canonical form of a matrix The matrix exponential Definition of the matrix exponential Computing the matrix exponential Graphs and eigenvalues Cospectral graphs Bipartite graphs and eigenvalues Regular graphs Distinct eigenvalues of a graph 330

5 xii Contents 6 Orthogonality and best approximation Norms and inner products Examples of norms and inner products The adjoint of a linear operator The adjoint of a linear operator Orthogonal vectors and bases Orthogonal bases The projection theorem Overdetermined linear systems The Gram-Schmidt process Least-squares polynomial approximation Orthogonal complements The fundamental theorem of linear algebra revisited Complex inner product spaces Examples of complex inner product spaces Orthogonality in complex inner product spaces The adjoint of a linear operator More on polynomial approximation A weighted L2 inner product The energy inner product and Galerkin's method Piecewise polynomials Continuous piecewise quadratic functions Higher degree finite element spaces Gaussian quadrature The trapezoidal rule and Simpson's rule Gaussian quadrature Orthogonal polynomials Weighted Gaussian quadrature The Helmholtz decomposition The divergence theorem Stokes's theorem The Helmholtz decomposition The spectral theory of symmetric matrices The spectral theorem for symmetric matrices Symmetric positive definite matrices Hermitian matrices The spectral theorem for normal matrices Outer products and the spectral decomposition Optimization and the Hessian matrix Background Optimization of quadratic functions Taylor's theorem First- and second-order optimality conditions Local quadratic approximations 446

6 Contents xiii 74 Lagrange multipliers Spectral methods for differential equations Eigenpairs of the differential operator Solving the BVP using eigenfunctions 456 The singular value decomposition Introduction to the SVD The SVD for singular matrices The SVD for general matrices Solving least-squares problems using the SVD The SVD and linear inverse problems Resolving inverse problems through regularization The truncated SVD method Tikhonov regularization The Smith normal form of a matrix An algorithm to compute the Smith normal form 852 Applications of the Smith normal form 501 Matrix factorizations and numerical linear algebra The LU factorization Operation counts Solving Ax = b using the LU factorization Partial pivoting Finite-precision arithmetic Examples of errors in Gaussian elimination Partial pivoting The PLU factorization The Cholesky factorization Matrix norms Examples of induced matrix norms The sensitivity of linear systems to errors Numerical stability Backward error analysis Analysis of Gaussian elimination with partial pivoting The sensitivity of the least-squares problem The QR factorization Solving the least-squares problem Computing the QR factorization Backward stability of the Householder QR algorithm Solving a linear system Eigenvalues and simultaneous iteration Reduction to triangular form The power method Simultaneous iteration The QR algorithm 572

7 xiv Contents 9101 A practical QR algorithm Reduction to upper Hessenberg form The explicitly shifted QR algorithm The implicitly shifted QR algorithm 579 LO Analysis in vector spaces Analysis 1011 Convergence and continuity in Rn Compactness 584 in R" Completeness of R" Equivalence of norms on R Infinite-dimensional vector spaces Banach and Hubert spaces Functional analysis The dual of a Hilbert space Weak convergence Convexity 611 A The Euclidean algorithm 617 A0-1 Computing multiplicative inverses in Zp 618 A02 Related results 619 B Permutations 621 C Polynomials 625 Cl Rings of polynomials 625 C2 Polynomial functions 630 C 21 Factorization of polynomials 632 D Summary of analysis in R 633 D 01 Convergence 633 D02 Completeness of R 634 D03 Open and closed sets 635 D04 Continuous functions 636 Bibliography 637 Index 641

Contents. Preface for the Instructor. Preface for the Student. xvii. Acknowledgments. 1 Vector Spaces 1 1.A R n and C n 2

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