Pairwise Comparisons Matrix Decomposition into Approximation and Orthogonal Component Using Lie Theory

TL;DR

This paper introduces a Lie theory-based matrix decomposition method for pairwise comparisons, separating approximation and orthogonal components to analyze inconsistency and improve decision-making accuracy.

Contribution

It develops a novel decomposition framework for pairwise comparison matrices using Lie group and Lie algebra theory, providing a theoretical basis for multiplicative comparisons.

Findings

Decomposition effectively separates approximation and orthogonal components.

Provides a new geometric perspective on inconsistency in pairwise comparisons.

Lays foundation for advanced inconsistency analysis in decision matrices.

Abstract

This paper examines the use of Lie group and Lie Algebra theory to construct the geometry of pairwise comparisons matrices. The Hadamard product (also known as coordinatewise, coordinate-wise, elementwise, or element-wise product) is analyzed in the context of inconsistency and inaccuracy by the decomposition method. The two designed components are the approximation and orthogonal components. The decomposition constitutes the theoretical foundation for the multiplicative pairwise comparisons. Keywords: approximate reasoning, subjectivity, inconsistency, consistency-driven, pairwise comparison, matrix Lie group, Lie algebra, approximation, orthogonality, decomposition.