A Note on Perturbations of Stochastic Matrices

Abstract

helmut wielandt zum 90. geburtstag Let A and B be stochastic matrices of the same type nn n. It is natural to consider perturbations At of A of the form At = =1 − tA + tB 0 ≤ t ≤ 1 Now, the following two questions come up immediately. (1) When does lim k→∞ At k exist? (2) How does lim k→∞ At k behave for small t? As the existence of lim k→∞ At k depends only on the eigenvalues of At, which are continuously dependent on t, question (1) allows a posi-tive answer (see Section 1). The answer to question (2) is less simple. This is not surprising, for if lim k→∞ At k =   v 1 t v n t   then v i tAt = v i t and it is well known that eigenvectors do not behave well under perturbations.