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Totally nonpositive completions on partial matrices

✍ Scribed by C. Mendes Araújo; Juan R. Torregrosa; Ana M. Urbano

Publisher
Elsevier Science
Year
2006
Tongue
English
Weight
183 KB
Volume
413
Category
Article
ISSN
0024-3795

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✦ Synopsis

An n × n real matrix is said to be totally nonpositive if every minor is nonpositive. In this paper, we are interested in totally nonpositive completion problems, that is, does a partial totally nonpositive matrix have a totally nonpositive matrix completion? This problem has, in general, a negative answer. Therefore, we analyze the question: for which labeled graphs G does every partial totally nonpositive matrix, whose associated graph is G, have a totally nonpositive completion? Here we study the mentioned problem when G is a chordal graph or an undirected cycle.

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two completion conjectures for partial upper triangular matrices. In this paper we show that one of them is not true in general, and we prove its validity for some particular cases. We also prove the equivalence between the two conjectures in the case of partial Hessenherg matrices.