Linear Algebra in Action: Third Edition

Linear algebra permeates mathematics, perhaps more so than any other single subject. It plays an essential role in pure and applied mathematics, statistics, computer science, and many aspects of physics and engineering. This book conveys in a user-friendly way the basic and advanced techniques of linear algebra from the point of view of a working analyst. The techniques are illustrated by a wide sample of applications and examples that are chosen to highlight the tools of the trade. In short, this is material that many of us wish we had been taught as graduate students. Roughly the first third of the book covers the basic material of a first course in linear algebra. The remaining chapters are devoted to applications drawn from vector calculus, numerical analysis, control theory, complex analysis, convexity and functional analysis. In particular, fixed point theorems, extremal problems, matrix equations, zero location and eigenvalue location problems, and matrices with nonnegative entries are discussed. Appendices on useful facts from analysis and supplementary information from complex function theory are also provided for the convenience of the reader. In this new edition, most of the chapters in the first edition have been revised, some extensively. The revisions include changes in a number of proofs, either to simplify the argument, to make the logic clearer or, on occasion, to sharpen the result. New introductory sections on linear programming, extreme points for polyhedra and a Nevanlinna-Pick interpolation problem have been added, as have some very short introductory sections on the mathematics behind Google, Drazin inverses, band inverses and applications of SVD together with a number of new exercises. Title Contents Preface to the third edition Preface to the second edition Preface to the first edition 1. Prerequisites 2. Dimension and rank 3. Gaussian elimination 4. Eigenvalues and eigenvectors 5. Towards the Jordan decomposition 6. The Jordan decomposition 7. Determinants 8. Companion matrices and circulants 9. Inequalities 10. Normed linear spaces 11. Inner product spaces 12. Orthogonality 13. Normal matrices 14. Projections, volumes, and traces 15. Singular value decomposition 16. Positive definite and semidefinite matrices 17. Determinants redux 18. Applications 19. Discrete dynamical systems 20. Continuous dynamical systems 21. Vector-valued functions 22. Fixed point theorems 23. The implicit function theorem 24. Extremal problems 25. Newton’s method 26. Matrices with nonnegative entries 27. Applications of matrices with nonnegative entries 28. Eigenvalues of Hermitian matrices 29. Singular values redux I 30. Singular values redux II 31. Approximation by unitary matrices 32. Linear functionals 33. A minimal norm problem 34. Conjugate gradients 35. Continuity of eigenvalues 36. Eigenvalue location problems 37. Matrix equations 38. A matrix completion problem 39. Minimal norm completions 40. The numerical range 41. Riccati equations 42. Supplementary topics 43. Toeplitz, Hankel, and de Branges Bibliography Notation index Subject index