A Note on the Proof of the Perron-Frobenius Theorem

Yun Cheng, Timothy Carson, Mohamed B. M. Elgindi

exas A&M University—Qatar, Doha, Qatar.
University of Chicago, Chicago, USA.
University of Texas, Austin, USA.

DOI: 10.4236/am.2012.311235                

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.

Keywords

Perron Eigenpair; Homotopy; Eigencurves; Positive Matrices; Interval Matrices

Conflicts of Interest

The authors declare no conflicts of interest.

References

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