1 Reachability cone of eventually exponentially nonnegative matrices

By Dimitrios Noutsos and Michael Tsatsomeros.

We examine the relation between eventual exponential nonnegativity of a matrix A (e^{tA} ≥ 0 for all sufficiently large t ≥ 0) and eventual nonnegativity of I + hA, h ≥ 0 ((I + hA)^{k} ≥ 0 for all sufficiently large k ≥ 0). As a consequence, we are able to characterize initial points x_{0} ∈ ℝ^{n} such that e^{tA}x_{0} becomes and remains nonnegative as exactly those points for which the discrete trajectories x^{(k)} = (I+hA)^{k}x_{0} become and remain nonnegative. This extents work on the reachability cone of exponentially nonnegative matrices by Neumann, Stern and Tsatsomeros [1]. [1] M. Neumann, R.J. Stern, and

M. Tsatsomeros. The reachability cones of essentially nonnegative matrices. Linear and Multilinear Algebra, 28:213–224, 1991.