Abstract. Linear differential systemsẋ(t) = Ax(t) (A ∈ R n×n , x 0 = x(0) ∈ R n , t ≥ 0) whose solutions become and remain nonnegative are studied. It is shown that the eigenvalue of A furthest to the right must be real and must possess nonnegative right and left eigenvectors. Moreover, for some a ≥ 0, A + aI must be eventually nonnegative, that is, its powers must become and remain entrywise nonnegative. Initial conditions x 0 that result in nonnegative states x(t) in finite time are shown to form a convex cone that is related to the matrix exponential e tA and its eventual nonnegativity.
Several necessary or sufficient conditions for a sign pattern to allow eventual positivity are established. It is also shown that certain families of sign patterns do not allow eventual positivity. These results are applied to show that for n ≥ 2, the minimum number of positive entries in an n×n sign pattern that allows eventual positivity is n+1, and to classify all 2×2 and 3×3 sign patterns as to whether or not the pattern allows eventual positivity. A 3 × 3 matrix is presented to demonstrate that the positive part of an eventually positive matrix need not be primitive, answering negatively a question of Johnson and Tarazaga.
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