Citation
Bradley, Gerald Lee (1966) On the Set of Eigenvalues of a Class of Equimodular Matrices. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/07K4DM12. https://resolver.caltech.edu/CaltechTHESIS:09172015130932781
Abstract
The structure of the set ϐ(A) of all eigenvalues of all complex matrices (elementwise) equimodular with a given n x n nonnegative matrix A is studied. The problem was suggested by O. Taussky and some aspects have been studied by R. S. Varga and B.W. Levinger.
If every matrix equimodular with A is nonsingular, then A is called regular. A new proof of the P. CamionA.J. Hoffman characterization of regular matrices is given.
The set ϐ(A) consists of m ≤ n closed annuli centered at the origin. Each gap, ɤ, in this set can be associated with a class of regular matrices with a (unique) permutation, π(ɤ). The association depends on both the combinatorial structure of A and the size of the a_{ii}. Let A be associated with the set of r permutations, π_{1}, π_{2},…, π_{r}, where each gap in ϐ(A) is associated with one of the π_{k}. Then r ≤ n, even when the complement of ϐ(A) has n+1 components. Further, if π(ɤ) is the identity, the real boundary points of ɤ are eigenvalues of real matrices equimodular with A. In particular, if A is essentially diagonally dominant, every real boundary point of ϐ(A) is an eigenvalues of a real matrix equimodular with A.
Several conjectures based on these results are made which if verified would constitute an extension of the PerronFrobenius Theorem, and an algebraic method is introduced which unites the study of regular matrices with that of ϐ(A).
Item Type:  Thesis (Dissertation (Ph.D.))  

Subject Keywords:  (Mathematics)  
Degree Grantor:  California Institute of Technology  
Division:  Physics, Mathematics and Astronomy  
Major Option:  Mathematics  
Thesis Availability:  Public (worldwide access)  
Research Advisor(s): 
 
Thesis Committee: 
 
Defense Date:  4 April 1966  
Funders: 
 
Record Number:  CaltechTHESIS:09172015130932781  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:09172015130932781  
DOI:  10.7907/07K4DM12  
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  9156  
Collection:  CaltechTHESIS  
Deposited By:  INVALID USER  
Deposited On:  17 Sep 2015 21:14  
Last Modified:  27 Feb 2024 20:37 
Thesis Files

PDF
 Final Version
See Usage Policy. 19MB 
Repository Staff Only: item control page