If M is a matrix with positive entries, then its largest eigenvalue λ is positive and simple; moreover, there exist vectors v and w with positive components such that vM = λv and Mw = λw, if the inner product of v with w is 1, then the limit of λ - n times the i, jth entry of Mⁿ as n goes to infinity is the product of the ith component of w and the jth component of v.