TY - JOUR

T1 - Sets of Parter vertices which are Parter sets

AU - Fernandes, Maria do Rosário Silva Franco

AU - da Cruz, Henrique F.

N1 - SCOPUSID:84896897248
WOS:000336695400004

PY - 2014/5/1

Y1 - 2014/5/1

N2 - Given. an Hermitian matrix, whose graph is a tree, having a multiple eigenvalue lambda, the Parter-Wiener theorem guarantees the existence of principal submatrices for which the multiplicity of lambda increases. The vertices of the tree whose removal gives rise to these principal submatrices are called weak Parter vertices and with some additional conditions are called Parter vertices. A set of k Parter vertices whose removal increases the multiplicity of lambda by k is called Parter set. As observed by several authors a set of Parter vertices is not necessarily a Parter set. In this paper we prove that if A is a symmetric matrix, whose graph is a tree, and lambda is an eigenvalue of A whose multiplicity does not exceed 3, then every set of Parter vertices, for lambda relative to A, is also a Parter set.

AB - Given. an Hermitian matrix, whose graph is a tree, having a multiple eigenvalue lambda, the Parter-Wiener theorem guarantees the existence of principal submatrices for which the multiplicity of lambda increases. The vertices of the tree whose removal gives rise to these principal submatrices are called weak Parter vertices and with some additional conditions are called Parter vertices. A set of k Parter vertices whose removal increases the multiplicity of lambda by k is called Parter set. As observed by several authors a set of Parter vertices is not necessarily a Parter set. In this paper we prove that if A is a symmetric matrix, whose graph is a tree, and lambda is an eigenvalue of A whose multiplicity does not exceed 3, then every set of Parter vertices, for lambda relative to A, is also a Parter set.

KW - Eigenvalues

KW - Parter set

KW - Parter vertices

KW - Tree

U2 - 10.1016/j.laa.2014.02.004

DO - 10.1016/j.laa.2014.02.004

M3 - Article

SN - 0024-3795

VL - 448

SP - 37

EP - 54

JO - Linear Algebra and its Applications

JF - Linear Algebra and its Applications

ER -