# Special cases of pairwise comparisons matrices represented by Toeplitz matrices

## TL;DR

This paper investigates special Toeplitz pairwise comparison matrices with three entries, analyzing their eigenvalue-based inconsistency index and introducing a new circulant matrix class, providing exact formulas and estimates.

## Contribution

It introduces a new class of circulant pairwise comparison matrices and analyzes their inconsistency indices using mathematical methods.

## Key findings

- Eigenvalue-based inconsistency index varies across studied matrices.
- Exact formulas and estimations for inconsistency indices are provided.
- The new circulant matrix class covers a broad range of inconsistency levels.

## Abstract

This study presents special cases of inconsistent pairwise comparisons PC matrices and analysis of their eigenvalue-based inconsistency index using mathematical methods. All studied special cases of PC matrices are Toeplitz matrices with only three different entries $1$, $x$, and $1/x$. A new type of circulant pairwise comparisons matrix has been introduced. Although this class of PC matrices may be perceived as restricted, it is general enough to cover numerous levels of eigenvalue-based inconsistency index from the lowest to the highest. Both exact mathematical expressions and estimations, where the exact expression was impossible to find, are provided