Eventually Nonnegative Matrices and their Sign Patterns

A matrix A ∈ Rn×n is eventually nonnegative (respectively, eventually positive) if there exists a positive integer k0 such that for all k ≥ k0, A ≥ 0 (respectively, A > 0). Here inequalities are entrywise and all matrices are real and square. An eigenvalue of A is dominant if its magnitude is equal to the spectral radius of A. A matrix A has the strong Perron-Frobenius property if A has a unique dominant eigenvalue that is positive, simple, and has a positive eigenvector. It is well known (see, e.g., [10]) that the set of matrices for which both A and A have the strong Perron-Frobenius property coincides with the set of eventually positive matrices. Eventually nonnegative matrices and eventually positive matrices have applications to positive control theory (see, e.g., [13]). A sign pattern (matrix) is a matrix having entries in {+,−, 0}. For a real matrix A, sgn(A) is the sign pattern having entries that are the signs of the corresponding entries in A. The idea of studying sign patterns was introduced by the economist Paul Samuelson to model certain problems in economics for which the signs (but not the magnitudes) of the matrix entries are known. If A is an n× n sign pattern, the sign pattern class of A, denoted Q(A), is the set of all A ∈ Rn×n such that sgn(A) = A. If P is a property of a real matrix, then a sign pattern A requires P if every real matrix A ∈ Q(A) has property P , and A allows P or is potentially P if there is some A ∈ Q(A) that has property P . Numerous properties have been investigated from the point of view of characterizing sign patterns that require or allow a particular property (see, e.g, [5, 9] and the references therein). Sign patterns that require eventual positivity or eventual nonnegativity are characterized in [7]. Potentially eventually positive (PEP) sign patterns are studied in [1], where several necessary or sufficient conditions are given for a sign pattern to be PEP, and PEP sign patterns of order at most three are characterized. Much less is known about whether a sign pattern is potentially eventually nonnegative (PEN) as compared with whether it is PEP, although there have been numerous papers on eventually nonnegative matrices (see for example [2, 3, 6, 8, 11, 12, 13, 14]).