Efficient Vectors for Block Perturbed Consistent Matrices

IF 1.5 2区数学 Q2 MATHEMATICS, APPLIED

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-02-08 DOI:10.1137/23m1580310

Susana Furtado, Charles Johnson

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引用次数: 0

Abstract

SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 1, Page 601-618, March 2024.
Abstract. 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.

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块扰动一致矩阵的高效向量

SIAM 矩阵分析与应用期刊》，第 45 卷，第 1 期，第 601-618 页，2024 年 3 月。摘要在基于成对比较的优先级排序方案（如分析层次过程）中，从倒易矩阵中提取一个不可能一致的心排序向量非常重要。从有效的向量中选择这样一个向量是很自然的。最近，人们发现了一种方法，可以归纳生成任何倒易矩阵的所有有效向量。在这里，我们重点研究倒易矩阵的有效向量，倒易矩阵是一致矩阵的块扰动，即它是由一致矩阵通过只修改适当的主子矩阵中的条目得到的。我们为这类矩阵确定了一类明确的有效向量。在此基础上，我们给出了 3 乘 3 块扰动情况下所有高效向量的描述。此外，我们还给出了此类矩阵的右佩伦特征向量有效的充分条件，并举例说明了不存在有效特征向量的情况。此外，我们还考虑了某类恒定块扰动一致矩阵，对于这类矩阵，我们可以构建一类高效向量，并证明 Perron 特征向量的高效性。我们还提供了适当的例子。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Matrix Analysis and Applications 数学-应用数学

CiteScore

2.90

自引率

6.70%

发文量

审稿时长

6-12 weeks

期刊介绍： The SIAM Journal on Matrix Analysis and Applications contains research articles in matrix analysis and its applications and papers of interest to the numerical linear algebra community. Applications include such areas as signal processing, systems and control theory, statistics, Markov chains, and mathematical biology. Also contains papers that are of a theoretical nature but have a possible impact on applications.