Linear Algebra and Matrices: Topics for a Second Course

Description

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. 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