Perron–Frobenius theorem for matrices with some negative entries

On general matrices having the Perron-Frobenius Property

“…In [19], the Perron-Frobenius property stands for having the spectral radius as a positive eigenvalue with a nonnegative eigenvector; this is also the definition used in [6]. On the other hand, in this paper we say that a real matrix A possesses the PerronFrobenius property if ρ(A) (whether it is zero or positive) is an eigenvalue of A having a nonnegative eigenvector (which is the same as the definition introduced in [29]). Moreover, we say that A possesses the strong Perron-Frobenius property if A has a simple, positive, and strictly dominant eigenvalue with a positive eigenvector (which is the same as the definition introduced in [19]).…”

“…They called such a property the Perron property. Other earlier papers looking at issues relating to the spectral radius being an eigenvalue, at positive or nonnegative corresponding eigenvector, or at matrices with these properties, include [11], [12], [13], [16], [17], [19], [26], [27], [29].…”

On complex power nonnegative matrices

“…Real eventually nonnegative (positive) matrices were introduced by Friedland [5]. Such matrices have been widely studied and, in particular, several recent works aimed at extending some classical results of the Perron-Frobenius theory for nonnegative matrices, to eventually nonnegative matrices, see for instance [9][10][11][12]15,17]. Is it possible to do the same for the more general power nonnegative matrices?…”

Two characterizations of matrices with the Perron–Frobenius property

“…There is also interest in general matrices (which may not be eventually nonnegative) having the Perron-Frobenius property; see, e.g., [6], [10], [12], [14], [16], [21]. Splittings for these matrices were studied in [5], [16], [17].…”