On Orthogonal Projections on the Space of Consistent Pairwise Comparisons Matrices

TL;DR

This paper explores how orthogonal projections under different inner products affect the approximation of inconsistent pairwise comparison matrices, revealing that the choice of inner product influences the resulting consistent approximations.

Contribution

It introduces a method to derive priority vectors from pairwise comparison matrices considering various inner products, highlighting the dependence of approximations on the inner product used.

Findings

Orthogonalization properties vary with different inner products.

Approximate consistent matrices depend on the chosen inner product.

The method provides a new way to derive priority vectors considering inner product choice.

Abstract

In this study, the orthogonalization process for different inner products is applied to pairwise comparisons. Properties of consistent approximations of a given inconsistent pairwise comparisons matrix are examined. A method of a derivation of a priority vector induced by a pairwise comparison matrix for a given inner product has been introduced. The mathematical elegance of orthogonalization and its universal use in most applied sciences has been the motivating factor for this study. However, the finding of this study that approximations depend on the inner product assumed, is of considerable importance.