On the elasticity of the Perron root of a nonnegative matrix

Let A = (a(i,j)) bean n x n nonnegative irreducible matrix whose Perron root is lambda. The quantity e(i,j) = a(i,j)/lambda partial derivativelambda/partial derivativea(i,j) is known as the elasticity of lambda with respect to a(i,j). In this paper, we give two proofs of the fact that partial derivativee(i,j)/partial derivativea(i,j) greater than or equal to 0 so that e(i,j) is increasing as a function of a(i,j). One proof uses ideas from symbolic dynamics, while the other, which is matrix theoretic, also yields a characterization of the case when partial derivativee(i,j)/partial derivativea(i,j) = 0. We discuss a resulting connection between the elements of A and the elements of the group inverse of lambdaI-A.