2025/01/17 by Waldemar W. Koczkodaj, Witold Pedrycz, Koczkodaj, Waldemar W. +7
Computer Science · Mathematics · #53B50 #53Z50 #Differential Geometry (math.DG) #FOS: Mathematics #Mathematical Inequalities and Applications #Matrix Theory and Algorithms
paper · pdf · doi:10.48550/arxiv.2501.10014
openalex publication_date 2025/01/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Symmetry and antisymmetry are fundamental concepts in many strict sciences. Pairwise comparisons (PC) matrices are fundamental tools for representing pairwise relations in decision making. In this theoretical study, we present a novel framework that embeds additive skew-symmetric PC matrices into the Grassmannian manifold G(2, n). This framework leverages Plücker coordinates to provide a rigorous geometric interpretation of their structure. Our key result demonstrates that the algebraic consistency condition aij + ajk - aki = 0 is equivalent to the geometric consistency of 2-planes in G(2, n), satisfying the Plücker relations. This connection reveals that the algebraic properties of PC matrices can be naturally understood through their geometric representation. Additionally, we extend this framework by interpreting PC matrices as differential 2-forms, providing a new perspective on their consistency as a closedness condition. Our framework of linear algebra, differential geometry, and algebraic geometry, placing PC matrices in a broader mathematical context. Rather than proposing a practical alternative to existing methods, our study aims to offer a theoretical foundation for future research by exploring new insights into higher-dimensional geometry and the geometric modeling of pairwise comparisons.