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Refined inertially and spectrally arbitrary zero-nonzero patterns

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.