Geodesic
Refined inertially and spectrally arbitrary zero-nonzero patterns
The electronic journal of linear algebra, Tome 20 (2010), pp. 449-467.

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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
Keywords: zero-nonzero pattern, refined inertia, inertially arbitrary, spectrally arbitrary 
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     author = {Deaett, Louis and Olesky, Dale D. and Den Driessche, Pauline Van},
     title = {Refined inertially and spectrally arbitrary zero-nonzero patterns},
     journal = {The electronic journal of linear algebra},
     pages = {449--467},
     publisher = {mathdoc},
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     year = {2010},
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Deaett, Louis; Olesky, Dale D.; Den Driessche, Pauline Van. Refined inertially and spectrally arbitrary zero-nonzero patterns. The electronic journal of linear algebra, Tome 20 (2010), pp. 449-467. http://geodesic.mathdoc.fr/item/ELA_2010__20__a20/