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
SN - 0024-3795
VL - 559
SP - 1
EP - 10
JO - Linear Algebra and its Applications
JF - Linear Algebra and its Applications
ER -