On the Elasticity of the Perron Root of a Nonnegative Matrix

Abstract

Let A=(ai,j) be an n × n nonnegative irreducible matrix whose Perron root is $\lambda$. The quantity $e_{i,j}=\frac{a_{i,j}}{\lam}\frac{\partial \lambda}{\partial a_{i,j}}$ is known as the elasticity of $\lambda$ with respect to ai,j. In this paper, we give two proofs of the fact that $\frac{\partial e_{i,j}}{\partial a_{i,j}} \geq 0$ so that ei,j is increasing as a function of ai,j. One proof uses ideas from symbolic dynamics, while the other, which is matrix theoretic, also yields a characterization of the case when $\frac{\partial e_{i,j}}{\partial a_{i,j}}=0$. We discuss a resulting connection between the elements of A and the elements of the group inverse of $\lambda I -A.$