Matrix Analysis and Applied Linear Algebra

Matrix Analysis and Applied Linear Algebra is an honest math text that circumvents the traditional definition-theorem-proof format that has bored students in the past. Meyer uses a fresh approach to introduce a variety of problems and examples ranging from the elementary to the challenging and from simple applications to discovery problems. The focus on applications is a big difference between this book and others. Meyer's book is more rigorous and goes into more depth than some. He includes some of the more contemporary topics of applied linear algebra which are not normally found in undergraduate textbooks. Modern concepts and notation are used to introduce the various aspects of linear equations, leading readers easily to numerical computations and applications. The theoretical developments are always accompanied with examples, which are worked out in detail. Each section ends with a large number of carefully chosen exercises from which the students can gain further insight.The textbook contains more than 240 examples, 650 exercises, historical notes, and comments on numerical performance and some of the possible pitfalls of algorithms. It comes with a solutions manual that includes complete solutions to all of the exercises. As a bonus, a CD-ROM is included that contains a searchable copy of the entire textbook and all solutions. Detailed information on topics mentioned in examples, references for additional study, thumbnail sketches and photographs of mathematicians, and a history of linear algebra.Students will love the book's clear presentation and informal writing style. The detailed applications are valuable to them in seeing how linear algebra is applied to real-life situations. One of the most interesting aspects of this book, however, is the inclusion of historical information. These personal insights into some of the greatest mathematicians who developed this subject provide a spark for students and make the teaching of this topic more fun. Table of Contents......Page 2 Preface......Page 6 Introduction......Page 10 Gaussian Elimination and Matrices......Page 12 Gauss-Jordan Method......Page 24 Two-Point Boundary Value Problems......Page 27 Making Gaussian Elimination Work......Page 30 Ill-Conditioned Systems......Page 42 Row Echelon Form and Rank......Page 50 Reduced Row Echelon Form......Page 56 Consistency of Linear Systems......Page 62 Homogeneous Systems......Page 66 Nonhomogeneous Systems......Page 73 Electrical Circuits......Page 82 From Ancient China to Arthur Cayley......Page 88 Addition and Transposition......Page 90 Linearity......Page 98 Why Do It This Way......Page 102 Matrix Multiplication......Page 104 Properties of Matrix Multiplication......Page 114 Matrix Inversion......Page 124 Inverses of Sums and Sensitivity......Page 133 Elementary Matrices and Equivalence......Page 140 The LU Factorization......Page 150 Spaces and Subspaces......Page 168 Four Fundamental Subspaces......Page 178 Linear Independence......Page 190 Basis and Dimension......Page 203 More About Rank......Page 219 Classical Least Squares......Page 232 Linear Transformations......Page 247 Change of Basis and Similarity......Page 260 Invariant Subspaces......Page 268 Vector Norms......Page 278 Matrix Norms......Page 288 Inner-Product Spaces......Page 295 Orthogonal Vectors......Page 303 Gram-Schmidt Procedure......Page 316 Unitary and Orthogonal Matrices......Page 329 Orthogonal Reduction......Page 350 Discrete Fourier Transform......Page 365 Complementary Subspaces......Page 392 Range-Nullspace Decomposition......Page 403 Orthogonal Decomposition......Page 412 Singular Value Decomposition......Page 420 Orthogonal Projection......Page 438 Why Least Squares?......Page 455 Angles Between Subspaces......Page 459 Determinants......Page 468 Additional Properties of Determinants......Page 484 Elementary Properties of Eigensystems......Page 498 Diagonalization by Similarity Transformations......Page 514 Functions of Diagonalizable Matrices......Page 534 Systems of Differential Equations......Page 550 Normal Matrices......Page 556 Positive Definite Matrices......Page 567 Nilpotent Matrices and Jordan Structure......Page 583 Jordan Form......Page 596 Functions of Nondiagonalizable Matrices......Page 608 Difference Equations, Limits, and Summability......Page 625 Minimum Polynomials and Krylov Methods......Page 651 Introduction......Page 670 Positive Matrices......Page 672 Nonnegative Matrices......Page 679 Stochastic Matrices and Markov Chains......Page 696 Index......Page 714 In this text, Meyer (mathematics, North Carolina State U.) circumvents the traditional definition-theorem-proof format by focusing on applications. He includes some of the more contemporary topics of applied linear algebra, uses modern concepts and notation, and accompanies theoretical developments with examples. The eight chapters cover linear equations, rectangular systems and echelon forms, matrix algebra, vector spaces, determinants, Eigenvalues and Eigenvectors, Perron-Frobenius theory, and norms, inner products, and orthogonality. The included CD-ROM contains a searchable copy of the entire textbook and all solutions, as well as detailed information on topics mentioned in examples, references, thumbnail sketches and photographs of mathematicians, and a history of linear algebra and computing.