A Note on the Proof of the Perron-Frobenius Theorem

• 作者: Mohamed B. M. Elgindi Timothy Carson Yun Cheng
• 来源期刊: 应用数学(英文)
• 学科: 数学
• 关键词: Perron Eigenpair HOMOTOPY Eigencurves POSITIVE MATRICES INTERVAL MATRICES
• 年，卷（期）: 2012,（11）
• 页码范围: 1697-1701
• 页数: 5页
• 分类号: O1

摘要

This paper provides a simple proof for the Perron-Frobenius theorem concerned with positive matrices using a homotopy technique. By analyzing the behaviour of the eigenvalues of a family of positive matrices, we observe that the conclusions of Perron-Frobenius theorem will hold if it holds for the starting matrix of this family. Based on our observations, we develop a simple numerical technique for approximating the Perron’s eigenpair of a given positive matrix. We apply the techniques introduced in the paper to approximate the Perron’s interval eigenvalue of a given positive interval matrix.