Potentially nilpotent sign pattern matrices

On block triangular matrices with signed Drazin inverse

“…For a square real matrix A, its sign pattern sgn A is called potentially nilpotent if there exists a nilpotent matrixà ∈ Q(A) (see [6]). It is known that the Drazin inverse of a nilpotent matrix A is always a zero matrix, and the nilpotent index of A equals its Drazin index ind(A).…”

On Nilpotence Indices of Sign Patterns

“…The nilpotence indices of realizations of a sign pattern have been studied in the literature (see, for example, [3,4,5]). However, the nilpotence index of a potentially nilpotent sign pattern is introduced in this paper for the first time.…”

“…The work in this paper was motivated by [3], where Eschenbach and Li listed four 4 by 4 sign patterns, conjectured to be nilpotent sign patterns of nilpotence index at least 3. These sign patterns with no zero entries, called full sign patterns, are shown to be potentially nilpotent of nilpotence index 3.…”

Spectrally arbitrary zero–nonzero patterns of order 4

“…Note that each spectrally arbitrary zero-nonzero (sign) pattern must allow nilpotency, must be inertially arbitrary, and must be potentially stable. These are three important sign pattern problems that are considered in the literature (see, for example, [3,6,7,8,9,10,12,13,14]). Note also that any spectrally arbitrary sign pattern must correspond to a signing of a spectrally arbitrary zero-nonzero pattern.…”