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Author(s)
António Leal Duarte; Charles R. Johnson;
Title Converse to the ParterWiener Theorem: the case of nontrees
Abstract Through a sucession of results, it is known that if the graph of an Hermitian matrix A is a tree and if for some index j, l Î s(A)Çs(A(j)), then there is an index i such that the multiplicity of l in s(A(i)) is one more than that in A. We exhibit a converse to this result by showing that it is generally true only for trees. In particular, it is shown that the minimum rank of a positive semidefinite matrix with a given graph G is £ n2 when G
is not a tree. This raises the question of how the minimum rank of a positive semidefinite matrix depends upon the graph in general.
Preprint series Prépublicações do Departamento de Matemática da Universidade de Coimbra
Issue 0331
Year 2003
