The Theory of Matrices: With Applications, Second Edition (Computer Science and Applied Mathematics)

In this book the authors try to bridge the gap between the treatments of matrix theory and linear algebra. It is aimed at graduate and advanced undergraduate students seeking a foundation in mathematics, computer science, or engineering. It will also be useful as a reference book for those working on matrices and linear algebra for use in their scientific work. Contents......Page f007.djvu Preface......Page f013.djvu 1 Matrix Algebra......Page p001.djvu 1.1 Special Types of Matrices ......Page p002.djvu 1.2 The Operations of Addition and Scalar Multiplication ......Page p004.djvu 1.3 Matrix Multiplication ......Page p007.djvu 1.4 Special Kinds of Matrices Related to Multiplication ......Page p010.djvu 1.5 Transpose and Conjugate Transpose ......Page p013.djvu 1.6 Submatrices and Partitions of a Matrix ......Page p016.djvu 1.7 Polynomials in a Matrix ......Page p019.djvu 1.8 Miscellaneous Exercises ......Page p021.djvu 2.1 Definition of the Determinant ......Page p023.djvu 2.2 Properties of Determinants ......Page p027.djvu 2.3 Cofactor Expansions ......Page p032.djvu 2.4 Laplace's Theorem ......Page p036.djvu 2.5 The Binet-Cauchy Formula ......Page p039.djvu 2.6 Adjoint and Inverse Matrices ......Page p042.djvu 2.7 Elementary Operations on Matrices ......Page p047.djvu 2.8 Rank of a Matrix ......Page p053.djvu 2.9 Systems of Linear Equations and Matrices ......Page p056.djvu 2.10 The LU Decomposition ......Page p061.djvu 2.11 Miscellaneous Exercises ......Page p063.djvu 3.1 Definition of a Linear Space ......Page p071.djvu 3.2 Subspaces ......Page p075.djvu 3.3 Linear Combinations ......Page p078.djvu 3.4 Linear Dependence and Independence ......Page p080.djvu 3.5 The Notion of a Basis ......Page p083.djvu 3.6 Sum and Direct Sum of Subspaces ......Page p087.djvu 3.7 Matrix Representation and Rank ......Page p091.djvu 3.8 Some Properties of Matrices Related to Rank ......Page p095.djvu 3.9 Change of Basis and Transition Matrices ......Page p098.djvu 3.10 Solution of Equations ......Page p100.djvu 3.11 Unitary and Euclidean Spaces ......Page p104.djvu 3.12 Orthogonal Systems ......Page p107.djvu 3.13 Orthogonal Subspaces ......Page p111.djvu 3.14 Miscellaneous Exercises ......Page p113.djvu 4 Linear Transformations and Matrices......Page p116.djvu 4.1 Linear Transformations ......Page p117.djvu 4.2 Matrix Representation of Linear Transformations ......Page p122.djvu 4.3 Matrix Representations, Equivalence, and Similarity ......Page p127.djvu 4.4 Some Properties of Similar Matrices ......Page p131.djvu 4.5 Image and Kernel of a Linear Transformation ......Page p133.djvu 4.6 Invertible Transformations ......Page p138.djvu 4.7 Restrictions, Invariant Subspaces, and Direct Sums of Transformations ......Page p142.djvu 4.8 Direct Sums and Matrices ......Page p145.djvu 4.9 Eigenvalues and Eigenvectors of a Transformation ......Page p147.djvu 4.10 Eigenvalues and Eigenvectors of a Matrix ......Page p152.djvu 4.11 The Characteristic Polynomial ......Page p155.djvu 4.12 The Multiplicities of an Eigenvalue ......Page p159.djvu 4.13 First Applications to Differential Equations ......Page p161.djvu 4.14 Miscellaneous Exercises ......Page p164.djvu 5.1 Adjoint Transformations ......Page p168.djvu 5.2 Normal Transformations and Matrices ......Page p174.djvu 5.3 Hermitian, Skew-Hermitian, and Definite Matrices ......Page p178.djvu 5.4 Square Root of a Definite Matrix and Singular Values ......Page p180.djvu 5.5 Congruence and the Inertia of a Matrix ......Page p184.djvu 5.6 Unitary Matrices ......Page p188.djvu 5.7 Polar and Singular-Value Decompositions ......Page p190.djvu 5.8 Idempotent Matrices (Projectors) ......Page p194.djvu 5.9 Matrices over the Field of Real Numbers ......Page p200.djvu 5.10 Bilinear, Quadratic, and Hermitian Forms ......Page p202.djvu 5.11 Finding the Canonical Forms ......Page p205.djvu 5.12 The Theory of Small Oscillations ......Page p208.djvu 5.13 Admissible Pairs of Matrices ......Page p212.djvu 5.14 Miscellaneous Exercises ......Page p217.djvu 6 The Jordan Canonical Form: A Geometric Approach......Page p220.djvu 6.1 Annihilating Polynomials ......Page p221.djvu 6.2 Minimal Polynomials ......Page p224.djvu 6.3 Generalized Eigenspaces ......Page p229.djvu 6.4 The Structure of Generalized Eigenspaces ......Page p232.djvu 6.5 The Jordan Theorem ......Page p236.djvu 6.6 Parameters of a Jordan Matrix ......Page p239.djvu 6.7 The Real Jordan Form ......Page p242.djvu 6.8 Miscellaneous Exercises ......Page p244.djvu 7.1 The Notion of a Matrix Polynomial ......Page p246.djvu 7.2 Division of Matrix Polynomials ......Page p248.djvu 7.3 Elementary Operations and Equivalence ......Page p253.djvu 7.4 A Canonical Form for a Matrix Polynomial ......Page p256.djvu 7.5 Invariant Polynomials and the Smith Canonical Form ......Page p259.djvu 7.6 Similarity and the First Normal Form ......Page p262.djvu 7.7 Elementary Divisors ......Page p265.djvu 7.8 The Second Normal Form and the Jordan Normal Form ......Page p269.djvu 7.9 The Characteristic and Minimal Polynomials ......Page p271.djvu 7.10 The Smith Form: Differential and Difference Equations ......Page p274.djvu 7.11 Miscellaneous Exercises ......Page p278.djvu 8 The Variational Method......Page p282.djvu 8.1 Field of Values. Extremal Eigenvalues of a Hermitian Matrix ......Page p283.djvu 8.2 Courant-Fischer Theory and the Rayleigh Quotient ......Page p286.djvu 8.3 The Stationary Property of the Rayleigh Quotient ......Page p289.djvu 8.4 Problems with Constraints ......Page p290.djvu 8.5 The Rayleigh Theorem and Definite Matrices ......Page p294.djvu 8.6 The Jacobi-Gundelfinger-Frobenius Method ......Page p296.djvu 8.7 An Application of the Courant-Fischer Theory ......Page p300.djvu 8.8 Applications to the Theory of Small Vibrations ......Page p302.djvu 9 Functions of Matrices......Page p304.djvu 9.1 Functions Defined on the Spectrum of a Matrix ......Page p305.djvu 9.2 Interpolatory Polynomials ......Page p306.djvu 9.3 Definition of a Function of a Matrix ......Page p308.djvu 9.4 Properties of Functions of Matrices ......Page p310.djvu 9.5 Spectral Resolution of f(A) ......Page p314.djvu 9.6 Component Matrices and Invariant Subspaces ......Page p320.djvu 9.7 Further Properties of Functions of Matrices ......Page p322.djvu 9 8 Sequences and Series of Matrices ......Page p325.djvu 9.9 The Resolvent and the Cauchy Theorem for Matrices ......Page p329.djvu 9.10 Applications to Differential Equations ......Page p334.djvu 9.11 Observable and Controllable Systems ......Page p340.djvu 9.12 Miscellaneous Exercises ......Page p345.djvu 10.1 The Notion of a Norm ......Page p350.djvu 10.2 A Vector Norm as a Metric: Convergence ......Page p354.djvu 10.3 Matrix Norms ......Page p358.djvu 10.4 Induced Matrix Norms ......Page p362.djvu 10.5 Absolute Vector Norms and Lower Bounds of a Matrix ......Page p367.djvu 10.6 The Geršgorin Theorem ......Page p371.djvu 10.7 Geršgorin Disks and Irreducible Matrices ......Page p374.djvu 10.8 The Schur Theorem ......Page p377.djvu 10.9 Miscellaneous Exercises ......Page p380.djvu 11.1 Perturbations in the Solution of Linear Equations ......Page p383.djvu 11.2 Perturbations of the Eigenvalues of a Simple Matrix ......Page p387.djvu 11.3 Analytic Perturbations ......Page p391.djvu 11.4 Perturbation of the Component Matrices ......Page p393.djvu 11.5 Perturbation of an Unrepeated Eigenvalue ......Page p395.djvu 11.6 Evaluation of the Perturbation Coefficients ......Page p397.djvu 11.7 Perturbation of a Multiple Eigenvalue ......Page p399.djvu 12.1 The Notion of a Kronecker Product ......Page p406.djvu 12.2 Eigenvalues of Kronecker Products and Composite Matrices ......Page p411.djvu 12.3 Applications of the Kronecker Product to Matrix Equations ......Page p413.djvu 12.4 Commuting Matrices ......Page p416.djvu 12.5 Solutions of AX + XB = C ......Page p421.djvu 12.6 One-Sided Inverses ......Page p424.djvu 12.7 Generalized Inverses ......Page p428.djvu 12.8 The Moore-Penrose Inverse ......Page p432.djvu 12.9 The Best Approximate Solution of the Equation Ax = b ......Page p435.djvu 12.10 Miscellaneous Exercises ......Page p438.djvu 13.1 The Lyapunov Stability Theory and Its Extensions ......Page p441.djvu 13.2 Stability with Respect to the Unit Circle ......Page p451.djvu 13.3 The Bezoutian and the Resultant ......Page p454.djvu 13.4 The Hermite and the Routh-Hurwitz Theorems ......Page p461.djvu 13.5 The Schur-Cohn Theorem ......Page p466.djvu 13.6 Perturbations of a Real Polynomial ......Page p468.djvu 13.7 The Liénard-Chipart Criterion ......Page p470.djvu 13.8 The Markov Criterion ......Page p474.djvu 13.9 A Determinantal Version of the Routh-Hurwitz Theorem ......Page p478.djvu 13.10 The Cauchy Index and Its Applications ......Page p482.djvu 14 Matrix Polynomials......Page p489.djvu 14.1 Linearization of a Matrix Polynomial ......Page p490.djvu 14.2 Standard Triples and Pairs ......Page p493.djvu 14.3 The Structure of Jordan Triples ......Page p500.djvu 14.4 Applications to Differential Equations ......Page p506.djvu 14.5 General Solutions of Differential Equations ......Page p509.djvu 14.6 Difference Equations ......Page p512.djvu 14.7 A Representation Theorem ......Page p516.djvu 14.8 Multiples and Divisors ......Page p518.djvu 14.9 Solvents of Monic Matrix Polynomials ......Page p520.djvu 15 Nonnegative Matrices......Page p527.djvu 15.1 Irreducible Matrices ......Page p528.djvu 15.2 Nonnegative Matrices and Nonnegative Inverses ......Page p530.djvu 15.3 The Perron-Frobenius Theorem (I) ......Page p532.djvu 15.4 The Perron-Frobenius Theorem (II) ......Page p538.djvu 15.5 Reducible Matrices ......Page p543.djvu 15.6 Primitive and Imprimitive Matrices ......Page p544.djvu 15.7 Stochastic Matrices ......Page p547.djvu 15.8 Markov Chains ......Page p550.djvu 1 A Survey of Scalar Polynomials ......Page p553.djvu 2 Some Theorems and Notions from Analysis ......Page p557.djvu 3 Suggestions for Further Reading ......Page p560.djvu Index ......Page p563.djvu "In this book the authors try to bridge the gap between the treatments of matrix theory and linear algebra to be found in current textbooks and the mastery of these topics required to use and apply our subject matter in several important areas of application, as well as in mathematics itself. At the same time we present a treatment that is as self-contained as is reasonable possible, beginning with the most fundamental ideas and definitions. In order to accomplish this double purpose, the first few chapters include a complete treatment of material to be found in standard courses on matrices and linear algebra. This part includes development of a computational algebraic development (in the spirit of the first edition) and also development of the abstract methods of finite-dimensional linear spaces. Indeed, a balance is maintained through the book between the two powerful techniques of matrix algebra and the theory of linear spaces and transformations."--1st paragraph of preface.