Refined inertially and spectrally arbitrary zero-nonzero patterns
ELA. The Electronic Journal of Linear Algebra, Tome 20 (2010), pp. 449-467.

Voir la notice de l'article dans Electronic Library of Mathematics

Summary: The refined inertia of a matrix is a quadruple specifying its inertia and additionally the number of its eigenvalues equal to zero. Spectral properties, especially the refined inertias, of real matrices with a given zero-nonzero pattern are investigated. It is shown that every zero-nonzero refined inertially arbitrary pattern of order 4 or less is also spectrally arbitrary. Irreducible and reducible examples are presented to show that for higher orders this is not the case. A further example shows that two zero-nonzero patterns that are not refined inertially arbitrary can have a direct sum that is refined inertially arbitrary, paralleling a known result for inertially arbitrary patterns. Analogously, it is shown that the direct sum of two zero-nonzero patterns may be spectrally arbitrary even if neither summand is spectrally arbitrary.
Classification : 15B35, 15A18, 05C50
Mots-clés : zero-nonzero pattern, refined inertia, inertially arbitrary, spectrally arbitrary 
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     title = {Refined inertially and spectrally arbitrary zero-nonzero patterns},
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Deaett, Louis; Olesky, Dale D.; Den Driessche, Pauline Van. Refined inertially and spectrally arbitrary zero-nonzero patterns. ELA. The Electronic Journal of Linear Algebra, Tome 20 (2010), pp. 449-467. https://geodesic-test.mathdoc.fr/item/EEJLA_2010__20__a20/