Linear Algebra and Matrices: Topics for a Second Course

✍ Scribed by Helene Shapiro

• Publisher: American Mathematical Society
• Year: 2015
• Tongue: English
• Leaves: 338
• Series: Pure and Applied Undergraduate Texts 24
• Category: Library

✦ Synopsis

Linear algebra and matrix theory are fundamental tools for almost every area of mathematics, both pure and applied. This book combines coverage of core topics with an introduction to some areas in which linear algebra plays a key role, for example, block designs, directed graphs, error correcting codes, and linear dynamical systems. Notable features include a discussion of the Weyr characteristic and Weyr canonical forms, and their relationship to the better-known Jordan canonical form; the use of block cyclic matrices and directed graphs to prove Frobenius's theorem on the structure of the eigenvalues of a nonnegative, irreducible matrix; and the inclusion of such combinatorial topics as BIBDs, Hadamard matrices, and strongly regular graphs. Also included are McCoy's theorem about matrices with property P, the Bruck-Ryser-Chowla theorem on the existence of block designs, and an introduction to Markov chains. This book is intended for those who are familiar with the linear algebra covered in a typical first course and are interested in learning more advanced results.

✦ Table of Contents

Contents
Preface
Note to the Reader
1. Preliminaries
2. Inner Product Spaces and Orthogonality
3. Eigenvalues, Eigenvectors, Diagonalization, and Triangularization
4. The Jordan and Weyr Canonical Forms
5. Unitary Similarity and Normal Matrices
6. Hermitian Matrices
7. Vector and Matrix Norms
8. Some Matrix Factorizations
9. Field of Values
10. Simultaneous Triangularization
11. Circulant and Block Cycle Matrices
12. Matrices of Zeros and Ones
13. Block Designs
14. Hadamard Matrices
15. Graphs
16. Directed Graphs
17. Nonnegative Matrices
18. Error-Correcting Codes
19. Linear Dynamical Systems
Bibliography
Index