The minimum number of eigenvalues of multiplicity one in a diagonalizable matrix, over a field, whose graph is a tree

Charles R. Johnson, António Leal-Duarte, Carlos M. Saiago

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

It is known that an n-by-n Hermitian matrix, n≥2, whose graph is a tree necessarily has at least two eigenvalues (the largest and smallest, in particular) with multiplicity 1. Recently, much of the multiplicity theory, for eigenvalues of Hermitian matrices whose graph is a tree, has been generalized to geometric multiplicities of eigenvalues of matrices over a general field (whose graph is a tree). However, the two 1's fact does not generalize. Here, we give circumstances under which there are two 1's and give several examples (without two 1's) that limit our positive results.

Original languageEnglish
Pages (from-to)1-10
Number of pages10
JournalLinear Algebra and Its Applications
Volume559
DOIs
Publication statusPublished - 15 Dec 2018

Keywords

  • Combinatorially symmetric
  • Diagonalizable matrix
  • Eigenvalues of multiplicity 1
  • Graph of a matrix
  • Matrix over a field
  • Tree

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