Efficiency of any weighted geometric mean of the columns of a reciprocal matrix

We focus upon the reciprocal matrix component of the often discussed Analytic Hierarchy Process and its approximation by a consistent matrix formed from an efficient vector. It is known that a vector is efficient for a reciprocal matrix A if and only if a certain digraph is strongly connected. Here, we give a transparent and relatively simple matricial proof of this important result. Then we show that any Hadamard weighted geometric mean of the columns of a reciprocal matrix is efficient. This is a major generalization of the known result that the Hadamard geometric mean of all columns is efficient, and of the recent result that the geometric mean of any subset of the columns is efficient. (c) 2023 Elsevier Inc. All rights reserved.