Potentially eventually exponentially positive sign patterns of the specific form

Abstract

An n × n sign pattern A is said to be potentially eventually exponentially
positive (PEEP) if there exists some real matrix A with the same sign
pattern as A and some nonnegative integer t0 such that for all t ≥ t0,
e
tA =
X∞
k=0
t
kAk
k!
> 0.
Identifying the necessary and sufficient conditions for an n × n(n ≥ 4)
sign pattern to be PEEP and classifying all PEEP sign patterns are two
open problems. In this short note, we investigate the potential eventual
exponential positivity of one class of sign patterns A of the specific form.
It is shown that for the special class of sign patterns, A is PEEP if and only
if all the nonzero off-diagonal entries of A is positive, which classifies all its
PEEP sign patterns of the specific form. Our result indicates that there
exist exactly one minimal PEEP sign pattern in the class of sign patterns
of the specific form.