António Leal Duarte

The maximum multiplicity of an eigenvalue in a matrix whose graph is a tree

We study the maximum possible multiplicity of an eigenvalue of a matrix whose graph is a tree, expressing that maximum multiplicity in terms of certain parameters associated with the tree.

On the relative position of multiple eigenvalues in the spectrum of an Hermitian matrix with a given graph

For Hermitian matrices, whose graph is a given tree, the relationships among vertex degrees, multiple eigenvalues and the relative position of the underlying eigenvalue in the ordered spectrum are discussed in detail. In the process, certain aspects of special vertices, whose removal results in an increase in multiplicity are investigated.

Construction of acyclic matrices from spectral data

Abstract Given real numbers λ 1 ,…,λ n ,μ 1 ,…,μ n −1 with λ 1 ⩾μ 2 ⩾λ 2 ⩾ … ⩾μ n −1⩾ λ n , an integer i with 1⩽ i ⩽ n , and a tree Γ with n vertices, we prove the existence of a Hermitian acyclic matrix A =[ x rs ] with eigenvalues λ 1 ,…,λ n and x rs =0 whenever r ≠ s and { r , s } is not an edge of Γ and such that the principal submatrix of A obtained from A by deleting the i th row and column has eigenvalues μ 1 ,…,μ n −1 .

Inverse eigenvalue problems and lists of multiplicities of eigenvalues for matrices whose graph is a tree: the case of generalized stars and double generalized stars

We characterize the possible lists of ordered multiplicities among matrices whose graph is a generalized star (a tree in which at most one vertex has degree greater than 2) or a double generalized star. Here, the inverse eigenvalue problem (IEP) for symmetric matrices whose graph is a generalized star is settled. The answer is consistent with a conjecture that determination of the possible ordered multiplicities is equivalent to the IEP for a given tree. Moreover, a key spectral feature of the IEP in the case of generalized stars is shown to characterize them among trees.

On the possible multiplicities of the eigenvalues of a Hermitian matrix whose graph is a tree

We consider the general problem of determining which lists of multiplicities for the eigenvalues occur among Hermitian matrices the graph of whose off-diagonal entries is a given tree. Several restrictions are cited and a construction strategy is given. Together, these are sufficient to characterize all lists for each tree in two infinite classes: the double paths and generalized stars, and to tabulate all lists for trees on fewer than nine vertices. Such tables should be useful for formulating and dispelling general conjectures.

Converse to the Parter-Wiener theorem: The case of non-trees

Through a succession of results, it is known that if the graph of an Hermitian matrix A is a tree and if for some index j, @l@?@s(A)@?@s(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 =

On the cartesian decomposition of a matrix

The main result of this paper is the following Let A and B be n×n hermitian matrices with eigenvalues respectively, ordered so that and let M1 be any k×k principal submatrix of . Necessary and sufficient conditions for equality are given.