TY - JOUR

T1 - The minimum number of eigenvalues of multiplicity one in a diagonalizable matrix, over a field, whose graph is a tree

AU - Johnson, Charles R.

AU - Leal-Duarte, António

AU - Saiago, Carlos M.

N1 - info:eu-repo/grantAgreement/FCT/5876/147204/PT#
Sem PDf conforme despacho.

PY - 2018/12/15

Y1 - 2018/12/15

N2 - 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.

AB - 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.

KW - Combinatorially symmetric

KW - Diagonalizable matrix

KW - Eigenvalues of multiplicity 1

KW - Graph of a matrix

KW - Matrix over a field

KW - Tree

UR - http://www.scopus.com/inward/record.url?scp=85052871176&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2018.08.033

DO - 10.1016/j.laa.2018.08.033

M3 - Article

AN - SCOPUS:85052871176

VL - 559

SP - 1

EP - 10

JO - Linear Algebra and its Applications

JF - Linear Algebra and its Applications

SN - 0024-3795

ER -