Geometric mapping properties of semipositive matrices

Semipositive matrices map a positive vector to a positive vector and as such they are a very broad generalization of the irreducible nonnegative matrices. Nevertheless, the ensuing geometric mapping properties of semipositive matrices result in several parallels to the theory of cone preserving and cone mapping matrices. It is shown that for a semipositive matrix A, there exist a proper polyhedral cone K1 of nonnegative vectors and a polyhedral cone K2 of nonnegative vectors such that AK1=K2. The set of all nonnegative vectors mapped by A to the nonnegative orthant is a proper polyhedral cone; as a consequence, A belongs to a proper polyhedral cone comprising semipositive matrices. When the powers Ak (k=0,1,…) have a common semipositivity vector, then A has a positive eigenvalue. If A has a sole peripheral eigenvalue λ and the powers of A have a common semipositivity vector with a non-vanishing term in the direction of the left eigenspace of λ, then A leaves a proper cone invariant.