Introduction To Matrix Theory: With Applications In Economics And Engineering (second Edition) (Series On Concrete And Applicable Mathematics)

Linear algebra and matrix theory are among the most important and most frequently applied branches of mathematics. They are especially important in solving engineering and economic models, where either the model is assumed linear, or the nonlinear model is approximated by a linear model, and the resulting linear model is examined. This book is mainly a textbook, that covers a one semester upper division course or a two semester lower division course on the subject. The second edition will be an extended and modernized version of the first edition. We added some new theoretical topics and some new applications from fields other than economics. We also added more difficult exercises at the end of each chapter which require deep understanding of the theoretical issues. We also modernized some proofs in the theoretical discussions which give better overview of the study material. In preparing the manuscript we also corrected the typos and errors, so the second edition will be a corrected, extended and modernized new version of the first edition. -- Provided by publisher Contents 10 Preface 6 List of Tables 14 List of Figures 16 Chapter 1 Vectors and Matrices 18 1.1 Introduction 18 1.2 Comparison of Matrices 24 1.3 Elementary Matrix Algebra 25 1.4 Inverse of a Matrix 38 1.5 Further Examples and Applications 41 1.6 Exercises 69 Chapter 2 Vector Spaces and Inner-Product Spaces 74 2.1 Introduction 74 2.2 Subspaces 78 2.3 Linear Independence, Basis 85 2.4 Inner-Product Spaces 97 2.5 Direct Sums and Orthogonal Complementary Subspaces 114 2.6 Applications 124 2.7 Exercises 131 Chapter 3 Systems of Linear Equations, and Inverses of Matrices 138 3.1 Introduction 138 3.2 Existence and Uniqueness of a Solution 140 3.3 Systems of Homogeneous Linear Equations 146 3.4 Systems of Inhomogeneous Linear Equations 151 3.5 Rank of Matrices 156 3.6 Matrix Equations and Inverses of Matrices 158 3.7 The Elimination Method 165 3.8 Applications 188 3.9 Exercises 204 Chapter 4 Determinants 212 4.1 Introduction 212 4.2 Properties of Determinants 219 4.3 Cofactors and Expansion by Cofactors 228 4.4 Determinants and Systems of Linear Equations 233 4.5 Further Examples and Applications 238 4.6 Exercises 246 Chapter 5 Linear Mappings and Matrices 252 5.1 Introduction 252 5.2 Vector Coordinates 252 5.3 Linear Mappings 255 5.4 The Vector Space of Linear Mappings 267 5.5 Multiplication of Linear Mappings, and Inverses 270 5.6 Matrix Representations of Linear Mappings 279 5.7 Coordinates and Matrix Representation in a New Basis 286 5.8 Applications 291 5.9 Exercises 300 Chapter 6 Eigenvalues, Invariant Subspaces, Canonical Forms 306 6.1 Introduction 306 6.2 Basic Concepts 306 6.3 Matrix Polynomials 316 6.4 The Construction of Invariant Subspaces 325 6.5 Diagonal and Triangular Forms 330 6.6 The Jordan Canonical Form 339 6.7 Complexification 345 6.8 Applications 347 6.9 Exercises 361 Chapter 7 Special Matrices 366 7.1 Introduction 366 7.2 Diagonal, Tridiagonal, and Triangular Matrices 366 7.3 Idempotent and Nilpotent Matrices 376 7.4 Matrices in Inner Product Spaces 377 7.5 Definite Matrices 389 7.6 Nonnegative Matrices 395 7.7 Applications 407 7.8 Exercises 421 Chapter 8 Elements of Matrix Analysis 426 8.1 Introduction 426 8.2 Vector Norms 426 8.3 Matrix Norms 432 8.4 Applications 443 8.5 Exercises 459 Bibliography 464 Index 466