Spectrally and inertially arbitrary sign patterns

Spectrally arbitrary zero–nonzero patterns of order 4

“…The problem of classifying sign patterns of matrices that are spectrally arbitrary was first proposed by Drew, Johnson, Olesky and van den Driessche [3] in 2000. Since that time there have been several papers on this topic (see for example , [1], [2], [4], [5], [8], [11], [12]). …”

“…In [2] it is shown that the graph of every spectrally arbitrary pattern must contain at least one 2-cycle. The pattern represented by the second matrix in the first row of Appendix B only has two cycles of length 2 or greater.…”

“…In [2], it is shown that there are inertially arbitrary sign patterns that are not spectrally arbitrary. Our Corollary 3.8 establishes that: Observation 1.9 There are inertially arbitrary 4 × 4 zero-nonzero patterns that are not spectrally arbitrary.…”

Inertially arbitrary nonzero patterns of order 4

“…characterized the spectrally arbitrary sign patterns of order 3. In [4] the inertially arbitrary sign patterns of order 3 were characterized and were shown to be identical to the spectrally arbitrary sign patterns of order 3. There are other recent papers which explore classes of spectrally and inertially arbitrary sign patterns (see for example [3,11,12]).…”

“…It was demonstrated in [4] that there is an order 4 sign pattern which is inertially but not spectrally arbitrary: we provide more order 4 sign patterns with this property in Section 3. In [2] it was noted that it is yet unknown whether every spectrally arbitrary nonzero pattern has a signing which is spectrally arbitrary.…”