The Theory of Matrices, Second Edition: With Applications (Computer Science and Scientific Computing) (Computer Science and Scientific Computing)

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 6 Preface......Page 12 1 Matrix Algebra......Page 16 1.1 Special Types of Matrices ......Page 17 1.2 The Operations of Addition and Scalar Multiplication ......Page 19 1.3 Matrix Multiplication ......Page 22 1.4 Special Kinds of Matrices Related to Multiplication ......Page 25 1.5 Transpose and Conjugate Transpose ......Page 28 1.6 Submatrices and Partitions of a Matrix ......Page 31 1.7 Polynomials in a Matrix ......Page 34 1.8 Miscellaneous Exercises ......Page 36 2.1 Definition of the Determinant ......Page 38 2.2 Properties of Determinants ......Page 42 2.3 Cofactor Expansions ......Page 47 2.4 Laplace's Theorem ......Page 51 2.5 The Binet-Cauchy Formula ......Page 54 2.6 Adjoint and Inverse Matrices ......Page 57 2.7 Elementary Operations on Matrices ......Page 62 2.8 Rank of a Matrix ......Page 68 2.9 Systems of Linear Equations and Matrices ......Page 71 2.10 The LU Decomposition ......Page 76 2.11 Miscellaneous Exercises ......Page 78 3.1 Definition of a Linear Space ......Page 86 3.2 Subspaces ......Page 90 3.3 Linear Combinations ......Page 93 3.4 Linear Dependence and Independence ......Page 95 3.5 The Notion of a Basis ......Page 98 3.6 Sum and Direct Sum of Subspaces ......Page 102 3.7 Matrix Representation and Rank ......Page 106 3.8 Some Properties of Matrices Related to Rank ......Page 110 3.9 Change of Basis and Transition Matrices ......Page 113 3.10 Solution of Equations ......Page 115 3.11 Unitary and Euclidean Spaces ......Page 119 3.12 Orthogonal Systems ......Page 122 3.13 Orthogonal Subspaces ......Page 126 3.14 Miscellaneous Exercises ......Page 128 4 Linear Transformations and Matrices......Page 131 4.1 Linear Transformations ......Page 132 4.2 Matrix Representation of Linear Transformations ......Page 137 4.3 Matrix Representations, Equivalence, and Similarity ......Page 142 4.4 Some Properties of Similar Matrices ......Page 146 4.5 Image and Kernel of a Linear Transformation ......Page 148 4.6 Invertible Transformations ......Page 153 4.7 Restrictions, Invariant Subspaces, and Direct Sums of Transformations ......Page 157 4.8 Direct Sums and Matrices ......Page 160 4.9 Eigenvalues and Eigenvectors of a Transformation ......Page 162 4.10 Eigenvalues and Eigenvectors of a Matrix ......Page 167 4.11 The Characteristic Polynomial ......Page 170 4.12 The Multiplicities of an Eigenvalue ......Page 174 4.13 First Applications to Differential Equations ......Page 176 4.14 Miscellaneous Exercises ......Page 179 5.1 Adjoint Transformations ......Page 183 5.2 Normal Transformations and Matrices ......Page 189 5.3 Hermitian, Skew-Hermitian, and Definite Matrices ......Page 193 5.4 Square Root of a Definite Matrix and Singular Values ......Page 195 5.5 Congruence and the Inertia of a Matrix ......Page 199 5.6 Unitary Matrices ......Page 203 5.7 Polar and Singular-Value Decompositions ......Page 205 5.8 Idempotent Matrices (Projectors) ......Page 209 5.9 Matrices over the Field of Real Numbers ......Page 215 5.10 Bilinear, Quadratic, and Hermitian Forms ......Page 217 5.11 Finding the Canonical Forms ......Page 220 5.12 The Theory of Small Oscillations ......Page 223 5.13 Admissible Pairs of Matrices ......Page 227 5.14 Miscellaneous Exercises ......Page 232 6 The Jordan Canonical Form: A Geometric Approach......Page 235 6.1 Annihilating Polynomials ......Page 236 6.2 Minimal Polynomials ......Page 239 6.3 Generalized Eigenspaces ......Page 244 6.4 The Structure of Generalized Eigenspaces ......Page 247 6.5 The Jordan Theorem ......Page 251 6.6 Parameters of a Jordan Matrix ......Page 254 6.7 The Real Jordan Form ......Page 257 6.8 Miscellaneous Exercises ......Page 259 7.1 The Notion of a Matrix Polynomial ......Page 261 7.2 Division of Matrix Polynomials ......Page 263 7.3 Elementary Operations and Equivalence ......Page 268 7.4 A Canonical Form for a Matrix Polynomial ......Page 271 7.5 Invariant Polynomials and the Smith Canonical Form ......Page 274 7.6 Similarity and the First Normal Form ......Page 277 7.7 Elementary Divisors ......Page 280 7.8 The Second Normal Form and the Jordan Normal Form ......Page 284 7.9 The Characteristic and Minimal Polynomials ......Page 286 7.10 The Smith Form: Differential and Difference Equations ......Page 289 7.11 Miscellaneous Exercises ......Page 293 8 The Variational Method......Page 297 8.1 Field of Values. Extremal Eigenvalues of a Hermitian Matrix ......Page 298 8.2 Courant-Fischer Theory and the Rayleigh Quotient ......Page 301 8.3 The Stationary Property of the Rayleigh Quotient ......Page 304 8.4 Problems with Constraints ......Page 305 8.5 The Rayleigh Theorem and Definite Matrices ......Page 309 8.6 The Jacobi-Gundelfinger-Frobenius Method ......Page 311 8.7 An Application of the Courant-Fischer Theory ......Page 315 8.8 Applications to the Theory of Small Vibrations ......Page 317 9 Functions of Matrices......Page 319 9.1 Functions Defined on the Spectrum of a Matrix ......Page 320 9.2 Interpolatory Polynomials ......Page 321 9.3 Definition of a Function of a Matrix ......Page 323 9.4 Properties of Functions of Matrices ......Page 325 9.5 Spectral Resolution of f(A) ......Page 329 9.6 Component Matrices and Invariant Subspaces ......Page 335 9.7 Further Properties of Functions of Matrices ......Page 337 9 8 Sequences and Series of Matrices ......Page 340 9.9 The Resolvent and the Cauchy Theorem for Matrices ......Page 344 9.10 Applications to Differential Equations ......Page 349 9.11 Observable and Controllable Systems ......Page 355 9.12 Miscellaneous Exercises ......Page 360 10.1 The Notion of a Norm ......Page 365 10.2 A Vector Norm as a Metric: Convergence ......Page 369 10.3 Matrix Norms ......Page 373 10.4 Induced Matrix Norms ......Page 377 10.5 Absolute Vector Norms and Lower Bounds of a Matrix ......Page 382 10.6 The Gerigorin Theorem ......Page 386 10.7 Gerlgorin Disks and Irreducible Matrices ......Page 389 10.8 The Schur Theorem ......Page 392 10.9 Miscellaneous Exercises ......Page 395 11.1 Perturbations in the Solution of Linear Equations ......Page 398 11.2 Perturbations of the Eigenvalues of a Simple Matrix ......Page 402 11.3 Analytic Perturbations ......Page 406 11.4 Perturbation of the Component Matrices ......Page 408 11.5 Perturbation of an Unrepeated Eigenvalue ......Page 410 11.6 Evaluation of the Perturbation Coefficients ......Page 412 11.7 Perturbation of a Multiple Eigenvalue ......Page 414 12.1 The Notion of a Kronecker Product ......Page 421 12.2 Eigenvalues of Kronecker Products and Composite Matrices ......Page 426 12.3 Applications of the Kronecker Product to Matrix Equations ......Page 428 12.4 Commuting Matrices ......Page 431 12.5 Solutions of AX + XB = C ......Page 436 12.6 One-Sided Inverses ......Page 439 12.7 Generalized Inverses ......Page 443 12.8 The Moore-Penrose Inverse ......Page 447 12.9 The Best Approximate Solution of the Equation Ax = b ......Page 450 12.10 Miscellaneous Exercises ......Page 453 13.1 The Lyapunov Stability Theory and Its Extensions ......Page 456 13.2 Stability with Respect to the Unit Circle ......Page 466 13.3 The Bezoutian and the Resultant ......Page 469 13.4 The Hermite and the Routh-Hurwitz Theorems ......Page 476 13.5 The Schur-Cohn Theorem ......Page 481 13.6 Perturbations of a Real Polynomial ......Page 483 13.7 The Li6nard-Chipart Criterion ......Page 485 13.8 The Markov Criterion ......Page 489 13.9 A Determinantal Version of the Routh-Hurwitz Theorem ......Page 493 13.10 The Cauchy Index and Its Applications ......Page 497 14 Matrix Polynomials......Page 504 14.1 Linearization of a Matrix Polynomial ......Page 505 14.2 Standard Triples and Pairs ......Page 508 14.3 The Structure of Jordan Triples ......Page 515 14.4 Applications to Differential Equations ......Page 521 14.5 General Solutions of Differential Equations ......Page 524 14.6 Difference Equations ......Page 527 14.7 A Representation Theorem ......Page 531 14.8 Multiples and Divisors ......Page 533 14.9 Solvents of Monic Matrix Polynomials ......Page 535 15 Nonnegative Matrices......Page 542 15.1 Irreducible Matrices ......Page 543 15.2 Nonnegative Matrices and Nonnegative Inverses ......Page 545 15.3 The Perron-Frobenius Theorem (I) ......Page 547 15.4 The Perron-Frobenius Theorem (II) ......Page 553 15.5 Reducible Matrices ......Page 558 15.6 Primitive and Imprimitive Matrices ......Page 559 15.7 Stochastic Matrices ......Page 556 15.8 Markov Chains ......Page 565 Appendix 1: A Survey of Scalar Polynomials ......Page 568 Appendix 2: Some Theorems and Notions from Analysis ......Page 572 Appendix 3: Suggestions for Further Reading ......Page 575 Index ......Page 578 "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. Matrix algebra; Determinants, inverse matrices, and rank; Linear, euclidean, and unitary spaces; Linear transformations and matrices; Linear transformations in unitary spaces and simple matrices; The jordan canonical form: a geometric approach; Matrix polynomials and normal forms; The variational method; Functions of matrices; Norms and bounds for eigenvalues; Perturbation theory; Linear matrices equations and generalized inverses; Stability problems; Matrix polynomials; Nonnegative matrices