The matrix exponential function can be used to solve systems of linear differential equations. For certain applications, it is of interest whether or not the matrix exponential function of a given matrix becomes and remains entrywise nonnegative after some time. Such matrices are called eventually exponentially nonnegative. Often the exact numerical entries in the matrix are not known (for example due to uncertainty in experimental measurements), but the qualitative information is usually known. In this dissertation we discuss what structure on the signs of the entries of a matrix guarantees that the matrix is eventually exponentially nonnegative.