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A note on the theoretical approach to Grassmannians and Plücker coordinates for additive skew-symmetric pairwise comparisons matrices

Waldemar W. Koczkodaj, Witold Pedrycz, Alexander Pigazzini, Laura P. Pigazzini, Richard Pincak

TL;DR

This work addresses the theoretical link between algebraic consistency of additive skew-symmetric PC matrices and its geometric counterpart on the Grassmannian $G(2,n)$ using Plücker coordinates. It develops an embedding of PC data as wedge products $v_i\wedge v_j$ and proves that $a_{ij}+a_{jk}-a_{ki}=0$ holds if and only if $(v_i\wedge v_j)+(v_j\wedge v_k)-(v_k\wedge v_i)=0$, with the associated Plücker coordinates satisfying the Plücker relations. The authors further interpret PC matrices as differential 2-forms $\omega=\sum p_{kl}\,dx_k\wedge dx_l$ where $\omega(v_i,v_j)=a_{ij}$ and connect the consistency condition to the closedness $d\omega=0$, offering a differential-geometric viewpoint. They also propose an inconsistency-reduction framework via minimizing $I_{geom}=\|\Delta_{geom}\|^2$ and a corresponding matrix $M$, while discussing degeneration scenarios; overall, the paper provides a theoretical foundation bridging linear algebra, differential geometry, and algebraic geometry for higher-dimensional geometric modeling of PC matrices.

Abstract

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 $a_{ij} + a_{jk} - a_{ki} = 0$ 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.

A note on the theoretical approach to Grassmannians and Plücker coordinates for additive skew-symmetric pairwise comparisons matrices

TL;DR

This work addresses the theoretical link between algebraic consistency of additive skew-symmetric PC matrices and its geometric counterpart on the Grassmannian using Plücker coordinates. It develops an embedding of PC data as wedge products and proves that holds if and only if , with the associated Plücker coordinates satisfying the Plücker relations. The authors further interpret PC matrices as differential 2-forms where and connect the consistency condition to the closedness , offering a differential-geometric viewpoint. They also propose an inconsistency-reduction framework via minimizing and a corresponding matrix , while discussing degeneration scenarios; overall, the paper provides a theoretical foundation bridging linear algebra, differential geometry, and algebraic geometry for higher-dimensional geometric modeling of PC matrices.

Abstract

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 . This framework leverages Plücker coordinates to provide a rigorous geometric interpretation of their structure. Our key result demonstrates that the algebraic consistency condition is equivalent to the geometric consistency of -planes in , 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 -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.
Paper Structure (10 sections, 1 theorem, 50 equations, 1 figure)

This paper contains 10 sections, 1 theorem, 50 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Mathematical foundations
  3. Skew-symmetric matrices
  4. Grassmannians and geometric representation of PC matrices
  5. Consistency of a skew-symmetric additive PC matrix and Plücker coordinates in the Grassmannian $G(2, n)$
  6. Relation between geometric and algebraic inconsistency
  7. Analysis of the construction of $M$ and implications for minimizing the inconsistency
  8. Management of M matrix degeneration cases
  9. Discussion and new perspectives for additive PC matrices
  10. Conclusion

Key Result

Theorem 3.1

Let $A = [a_{ij}]$ be a skew-symmetric additive PC matrix of order $n$, defined by $a_{ij} = s_i - s_j$, where $s_i = \|v_i\|^2 \in \mathbb{R}$ represents a scalar measure associated with option $i$. There exists a set of vectors $v_i = b_i e_i$, where $\{e_i\}_{i=1}^n$ is an orthonormal basis of $\

Figures (1)

  • Figure 1: Inconsistency in additive pairwise comparisons

Theorems & Definitions (2)

  • Theorem 3.1
  • proof