On the first and second order derivatives of the Perron vector

Abstract

In a previous work [5] the authors developed formulas for the second order partial derivatives of the Perron root as a function of the matrix entries at an essentially nonnegative and irreducible matrix. These formulas, which involve the group generalized inverse of an associated M-matrix, were used to investigate the concavity and convexity of the Perron root as a function of the entries. The authors now combine the above results together with an approach taken in an earlier joint paper [6] of the second author with L. Elsner and C. Johnson, and they develop formulas for the second order derivatives of an appropriately normalized Perron vector with respect to the matrix entries, which again are given in terms the group generalized inverse of an associated M-matrix. Convexity properties of the Perron vector as a function of the entries of the matrix are then examined. In addition, formulas for the first derivative of the Perron vector resulting from different normalizations of this eigenvector are also given. A by-product of one of these formulas yields that the group generalized inverse of a singular and irreducible M-matrix can be diagonally scaled to a matrix which is entrywise column diagonally dominant. © 1985.