About the BookAbout the AuthorTable of Contents

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.

Helene Shapiro, Department of Mathematics and Statistics, Swarthmore College, Swarthmore, PA

• Preface 12
• Note to the reader 16
• Preliminaries 18
• Inner product spaces and orthogonality 34
• Eigenvalues, eigenvectors, diagonalization, and triangularization 56
• The Jordan and Weyr canonical forms 68
• Unitary similarity and normal matrices 94
• Hermitian matrices 106
• Vector and matrix norms 130
• Some matrix factorizations 138
• Field of values 160
• Simultaneous triangularization 168
• Circulant and block cycle matrices 180
• Matrices of zeros and ones 186
• Block designs 202
• Hadamard matrices 224
• Graphs 238
• Directed graphs 252
• Nonnegative matrices 266
• Error-correcting codes 282
• Linear dynamical systems 296
• Bibliography 320
• Index 328