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On the Set of Eigenvalues of a Class of Equimodular Matrices


Bradley, Gerald Lee (1966) On the Set of Eigenvalues of a Class of Equimodular Matrices. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/07K4-DM12.


The structure of the set ϐ(A) of all eigenvalues of all complex matrices (elementwise) equimodular with a given n x n non-negative 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 non-singular, then A is called regular. A new proof of the P. Camion-A.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 aii. Let A be associated with the set of r permutations, π1, π2,…, πr, where each gap in ϐ(A) is associated with one of the πk. 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 Perron-Frobenius 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):
  • Todd, John (advisor)
  • Bohnenblust, Henri Frederic (advisor)
Thesis Committee:
  • Unknown, Unknown
Defense Date:4 April 1966
Funding AgencyGrant Number
Record Number:CaltechTHESIS:09172015-130932781
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9156
Deposited On:17 Sep 2015 21:14
Last Modified:27 Feb 2024 20:37

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