Geometric conditions for the existence of positive eigenvalues of matrices

AbstractFinite-dimensional theorems of Perron-Frobenius type are proved. For A∈Cnn and a nonnegative integer k, we let wk (A) be the cone generated by Ak, Ak+1,…in Cnn. We show that A satisfies the Perron-Schaefer condition if and only if the closure Wk(A) of wk(A) is a pointed cone. This theorem is closely related to several known results. If k⩾v0(A), the index of the eigenvalue 0 in spec A, we prove that A has a positive eigenvalue if and only if wk(A) is a pointed nonzero cone or, equivalently Wk(A) is not a real subspace of Cnn. Our proofs are elementary and based on a method of Birkhoff's. We discuss the relation of this method to Pringsheim's theorem