TITLE:
A Note on the Proof of the Perron-Frobenius Theorem
AUTHORS:
Yun Cheng, Timothy Carson, Mohamed B. M. Elgindi
KEYWORDS:
Perron Eigenpair; Homotopy; Eigencurves; Positive Matrices; Interval Matrices
JOURNAL NAME:
Applied Mathematics,
Vol.3 No.11,
November
20,
2012
ABSTRACT: 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.