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

摘要

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 l'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. (C) 2018 Elsevier Inc. All rights reserved.