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.
Author(s): Harry Dym
Series: Graduate Studies in Mathematics; 232
Edition: 3
Publisher: American Mathematical Society
Year: 2023
Language: English
Commentary: Fixed version of https://libgen.is/book/index.php?md5=0545112B846E20E60EE3F2EB811BCA71 (pages ordered correctly, ToC added)
Pages: 512
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
