EFFICIENT VECTORS FOR BLOCK PERTURBED CONSISTENT MATRICES

• Type: Article in International Scientific Journal
• Date: 2024

Authors

Susana Borges Furtado (Author) FEP; Johnson, C (Author) Other

Journal

Title: SIAM Journal on Matrix Analysis and Applications

Vol. 45

Pages: 601-618

ISSN: 0895-4798

Publisher: Society for Industrial and Applied Mathematics

Indexing

ISI Web of Knowledge - 0 Citations

Scopus - 5 Citations

Other information

Authenticus ID: P-010-2FQ

DOI: 10.1137/23m1580310

Abstract (EN): In prioritization schemes, based on pairwise comparisons, such as the analytical hierarchy process, it is important to extract a cardinal ranking vector from a reciprocal matrix that is unlikely to be consistent. It is natural to choose such a vector only from efficient ones. Recently, a method to generate inductively all efficient vectors for any reciprocal matrix has been discovered. Here we focus on the study of efficient vectors for a reciprocal matrix that is a block perturbation of a consistent matrix in the sense that it is obtained from a consistent matrix by modifying entries only in a proper principal submatrix. We determine an explicit class of efficient vectors for such matrices. Based on this, we give a description of all the efficient vectors in the 3 -by -3 block perturbed case. In addition, we give sufficient conditions for the right Perron eigenvector of such matrices to be efficient and provide examples in which efficiency does not occur. Also, we consider a certain type of constant block perturbed consistent matrices, for which we may construct a class of efficient vectors, and demonstrate the efficiency of the Perron eigenvector. Appropriate examples are provided throughout.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 18