The arrangement of nonzero entries of a matrix, described by the graph of the matrix, limits the possible geometric multiplicities of the eigenvalues, which are far more limited by this information than algebraic multiplicities or the numerical values of the eigenvalues. This book gives a unified development of how the graph of a symmetric matrix influences the possible multiplicities of its eigenvalues. While the theory is richest in cases where the graph is a tree, work on eigenvalues, multiplicities and graphs has provided the opportunity to identify which ideas have analogs for non-trees, and those for which trees are essential. It gathers and organizes the fundamental ideas to allow students and researchers to easily access and investigate the many interesting questions in the subject.
About the Author
Charles R. Johnson is Class of 1961 Professor of Mathematics at the College of William and Mary, Virginia. He is the recognized expert in the interplay between linear algebra and combinatorics, as well as many parts of matrix analysis. He is coauthor of Matrix Analysis (Cambridge, 2012), Topics in Matrix Analysis (Cambridge, 2010), both with Roger Horn, and Totally Nonnegative Matrices (2011, with Shaun Fallat).
Carlos M. Saiago is Assistant Professor of Mathematics at Universidade Nova de Lisboa, Portugal, and is the author of fifteen papers on eigenvalues, multiplicities, and graphs.
Table of Contents
Background; 1. Introduction; 2. Parter-Wiener, etc. theory; 3. Maximum multiplicity for trees, I; 4. Multiple eigenvalues and structure; 5. Maximum multiplicity, II; 6. The minimum number of distinct eigenvalues; 7. Construction techniques; 8. Multiplicity lists for generalized stars; 9. Double generalized stars; 10. Linear trees; 11. Non-trees; 12. Geometric multiplicities for general matrices over a field.