António Leal-Duarte - Academia.edu (original) (raw)
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Papers by António Leal-Duarte
For a given graph, there is a natural question of the possible lists of multiplicities for the ei... more For a given graph, there is a natural question of the possible lists of multiplicities for the eigenvalues among the spectra of Hermitian matrices with that graph (no constraint is placed upon the diagonal entries of the matrices by the graph). Here, we survey some of what is known about this question and include some new information about it. There is a natural focus upon the case in which the graph is a tree. In this event, there is remarkable structure to the possible lists. Both the general theory and a summary of specific results is given. At the end, this allows to give, in compact tabular form, all lists for trees on fewer than 11 vertices (a potentially valuable tool for further work). There is a brief discussion of non-trees.
For matrices over a field F, whose graph is a star, any characteristic polynomial may occur if |F... more For matrices over a field F, whose graph is a star, any characteristic polynomial may occur if |F| is large enough. Depending upon the diagonal entries, some linear factors will have to occur, but given this, the characteristic polynomial is still arbitrary. For smaller fields, a characterization of achievable polynomials is given. The geometrically multiple eigenvalues are easily identified, and, given this, the Jordan structure is completely determined. It turns out that no eigenvalue may enjoy more that one block of size greater than one, a restriction not present in all trees.
Linear and Multilinear Algebra, 1985
Let M be an arbitrary n x n complex matrix. As is well known, M may be written in a unique way in... more Let M be an arbitrary n x n complex matrix. As is well known, M may be written in a unique way in the form M = A + iB with A and B hermitian. (A and B are given by A = (M + M*)/2, B = (M - M*)/2i.) This is the so-called Cartesian deco~nposition of M. In this note we obtain ...
For a given graph, there is a natural question of the possible lists of multiplicities for the ei... more For a given graph, there is a natural question of the possible lists of multiplicities for the eigenvalues among the spectra of Hermitian matrices with that graph (no constraint is placed upon the diagonal entries of the matrices by the graph). Here, we survey some of what is known about this question and include some new information about it. There is a natural focus upon the case in which the graph is a tree. In this event, there is remarkable structure to the possible lists. Both the general theory and a summary of specific results is given. At the end, this allows to give, in compact tabular form, all lists for trees on fewer than 11 vertices (a potentially valuable tool for further work). There is a brief discussion of non-trees.
For matrices over a field F, whose graph is a star, any characteristic polynomial may occur if |F... more For matrices over a field F, whose graph is a star, any characteristic polynomial may occur if |F| is large enough. Depending upon the diagonal entries, some linear factors will have to occur, but given this, the characteristic polynomial is still arbitrary. For smaller fields, a characterization of achievable polynomials is given. The geometrically multiple eigenvalues are easily identified, and, given this, the Jordan structure is completely determined. It turns out that no eigenvalue may enjoy more that one block of size greater than one, a restriction not present in all trees.
Linear and Multilinear Algebra, 1985
Let M be an arbitrary n x n complex matrix. As is well known, M may be written in a unique way in... more Let M be an arbitrary n x n complex matrix. As is well known, M may be written in a unique way in the form M = A + iB with A and B hermitian. (A and B are given by A = (M + M*)/2, B = (M - M*)/2i.) This is the so-called Cartesian deco~nposition of M. In this note we obtain ...