Derivatives of the spectral radius as a function of non-negative matrix elements

Abstract

Let A = (a ij ) be a non-negative n × n matrix, that is, a ij ≥ 0, i, j = 1, …, n ; n > 1. We write A ≥ 0. Let r = r ( A ) be the spectral radius of A ; assume r > 0 throughout to avoid trivial cases. Let be the m th derivative of r with respect to the element a ij , all other elements of A being held constant.