Estudo Geralhttps://estudogeral.sib.uc.ptThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Sun, 25 Aug 2019 12:06:07 GMT2019-08-25T12:06:07Z5071Imbedding conditions for normal matriceshttp://hdl.handle.net/10316/10044Title: Imbedding conditions for normal matrices
Authors: Queiró, João Filipe; Duarte, António Leal
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/10316/100442009-01-01T00:00:00ZOn Fiedler's characterization of tridiagonal matrices over arbitrary fieldshttp://hdl.handle.net/10316/11425Title: On Fiedler's characterization of tridiagonal matrices over arbitrary fields
Authors: Bento, Américo; Duarte, António Leal
Abstract: M. Fiedler proved in [1] that the set of real n-by-n symmetric matrices A such that rank(A + D) ≥ n - 1 for every real diagonal matrix D is the set of matrices PT PT where P is a permutation matrix and T an irreducible tridiagonal matrix. We show that this result remains valid for arbitrary fields with some exceptions for 5-by-5 matrices over Z3
Wed, 01 Jan 2003 00:00:00 GMThttp://hdl.handle.net/10316/114252003-01-01T00:00:00ZInverse eigenvalue problems and lists of multiplicities of eigenvalues for matrices whose graph is a tree: the case of generalized stars and double generalized starshttp://hdl.handle.net/10316/11451Title: 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
Authors: Johnson, Charles R.; Duarte, António Leal; Saiago, Carlos M.
Abstract: 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 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 inverse eigenvalue problem for a given tree. Moreover,
a key spectral feature of the inverse eigenvalue problem in the
case of generalized stars is shown to characterize them among trees.
Tue, 01 Jan 2002 00:00:00 GMThttp://hdl.handle.net/10316/114512002-01-01T00:00:00ZThe change in eigenvalue multiplicity associated with perturbation of a diagonal entry of the matrixhttp://hdl.handle.net/10316/13630Title: The change in eigenvalue multiplicity associated with perturbation of a diagonal entry of the matrix
Authors: Johnson, Charles R.; Duarte, António Leal; Saiago, Carlos M.
Abstract: Here we investigate the relation between perturbing the i-th diagonal
entry of A 2 Mn(F) and extracting the principal submatrix A(i) from A with
respect to the possible changes in multiplicity of a given eigenvalue. A complete
description is given and used to both generalize and improve prior work about
Hermitian matrices whose graph is a given tree.
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/10316/136302009-01-01T00:00:00ZThe structure of matrices with a maximum multiplicity eigenvaluehttp://hdl.handle.net/10316/4579Title: The structure of matrices with a maximum multiplicity eigenvalue
Authors: Johnson, Charles R.; Duarte, António Leal; Saiago, Carlos M.
Abstract: There is remarkable and distinctive structure among Hermitian matrices, whose graph is a given tree T and that have an eigenvalue of multiplicity that is a maximum for T. Among such structure, we give several new results: (1) no vertex of T may be "neutral"; (2) neutral vertices may occur if the largest multiplicity is less than the maximum; (3) every Parter vertex has at least two downer branches; (4) removal of a Parter vertex changes the status of no other vertex; and (5) every set of Parter vertices forms a Parter set. Statements (3), (4) and (5) are also not generally true when the multiplicity is less than the maximum. Some of our results are used to give further insights into prior results, and both the review of necessary background and the development of new structural lemmas may be of independent interest.
Tue, 01 Jan 2008 00:00:00 GMThttp://hdl.handle.net/10316/45792008-01-01T00:00:00ZA Fiedler's type characterization of band matriceshttp://hdl.handle.net/10316/11402Title: A Fiedler's type characterization of band matrices
Authors: Bento, Américo; Duarte, António Leal
Abstract: Let K be a field and p an integer positive number. We denote by
Bpn
(K) the set of n-by-n symmetric band matrices of bandwidth 2p − 1, i.e., if
A = [aij ] ∈ Bpn
(K) then aij = 0 if |i − j| > p − 1. Let b Bpn
(K) be the set of matrices
from Bpn
(K) in which the entries (i, j), |i − j| = p − 1, are different from zero.
Let A be a n-by-n symmetric matrix with entries from K; and p such that 3 6
p 6 n. We will show that: rank(A + B) > n − p + 1, for every B ∈ Bp−1
n (K), if and
only if A ∈ b Bpn
(K).
Thu, 01 Jan 2004 00:00:00 GMThttp://hdl.handle.net/10316/114022004-01-01T00:00:00ZConverse to the Parter-Wiener Theorem: the case of non-treeshttp://hdl.handle.net/10316/11424Title: Converse to the Parter-Wiener Theorem: the case of non-trees
Authors: Johnson, Charles R.; Duarte, António Leal
Abstract: 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, λ e σ(A) ∩ σ(A(j)), then there is an index i such that the multiplicity of λ in σ(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
Wed, 01 Jan 2003 00:00:00 GMThttp://hdl.handle.net/10316/114242003-01-01T00:00:00Z