On The Eventual Exponential Positivity Of Some Tree Sign Patterns

An nxn matrix A is called eventually exponentially positive (EEP) if e(tA) (=) Sigma(infinity)(k=0)t(k)A(k)/k!>0 for all t >= t(0), where t(0) >= 0. A matrix whose entries belong to the set {+,-,0} is called a sign pattern. An nxn sign pattern A is called potentially eventually exponentially positive (PEEP) if there exists some real matrix realization A of A that is EEP. Characterizing the PEEP sign patterns is a longstanding open problem. In this article, A is called minimally potentially eventually exponentially positive (MPEEP), if A is PEEP and no proper subpattern of A is PEEP. Some preliminary results about MPEEP sign patterns and PEEP sign patterns are established. All MPEEP sign patterns of orders n <= 3 are identified. For the nxn tridiagonal sign patterns T-n, we show that there exists exactly one MPEEP tridiagonal sign pattern T-n(o). Consequently, we classify all PEEP tridiagonal sign patterns as the superpatterns of T-n(o). We also classify all PEEP star sign patterns S-n and double star sign patterns DS(n, m) by identifying all the MPEEP star sign patterns and the MPEEP double star sign patterns, respectively.