A Note on the Proof of the Perron-Frobenius Theorem

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

doi:10.4236/am.2012.311235

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

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

Cite this paper: Y. Cheng, T. Carson and M. Elgindi, "A Note on the Proof of the Perron-Frobenius Theorem," Applied Mathematics, Vol. 3 No. 11, 2012, pp. 1697-1701. doi: 10.4236/am.2012.311235.

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