## Hermitian matrices, eigenvalue multiplicities, and eigenvector components

Charles R. Johnson and Brian D. Sutton
*SIAM J. Matrix Anal. Appl.* 26 (2004/05), no. 2, 390–399.

Given an n-by-n Hermitian matrix A and a real number lambda, index i
is said to be Parter (resp. neutral, downer) if the multiplicity of lambda as an
eigenvalue of A(i) is one more (resp. the same, one less) than that in
A. In case the multiplicity of lambda in A is at least 2 and the graph of A is a
tree, there are always Parter vertices. Our purpose here is to advance
the classification of vertices and, in particular, to relate classification
to the combinatorial structure of eigenspaces. Some general results are
given and then used to deduce some rather specific facts, not otherwise
easily observed. Examples are given.

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