Consistent projections and indicators in pairwise comparisons

Abstract

This study examines several generic properties of weighted consistent projections and indicators of inconsistency in an arbitrary finite dimensional inner product space of square matrices. In the case of weighted Frobenius inner products we present explicit formulae for them in terms of the matrix entries and weights. It extends the recent results, due to Koczkodaj et al. [Fund. Inform. 172 (2020) 379–397], to positive matrices in pairwise comparisons. Moreover, we discuss the possibilities of a proper choice of the inner product weights for numerical applications based on the orthogonal consistent projections.