António Leal Duarte; Charles R. Johnson;
Converse to the Parter-Wiener Theorem: the case of non-trees
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 £ n-2 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.
Pré-publicações do Departamento de Matemática da Universidade de Coimbra