Hermitian Matrices, Eigenvalue Multiplicities, and Eigenvector Components ∗ Charles R. Johnson † , Brian D. Sutton ‡ July 25, 2002 Abstract Given an n-by-n Hermitian matrix A and a real number λ, index i is said to be Parter (resp. neutral, downer) if the multiplicity of λ as an eigenvalue of A(i) is one more (resp. the same, one less) than that in A. In case the multiplicity of λ 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. 1 Introduction Throughout, A will be an n-by-n Hermitian matrix and A(i) its (n − 1)- by-(n − 1) principal submatrix, resulting from deletion of row and column i, i =1,...,n. If λ ∈ R is an identified eigenvalue, we denote by m A (λ) * Research supported by NSF-REU grant DMS-99-87803. † Department of Mathematics, College of William and Mary, Williamsburg, VA 23187, USA ‡ Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139–4307, USA. Research supported by a National Science Foundation Graduate Re- search Fellowship. 1