A Note on the Proof of the Perron-Frobenius Theorem
Affiliation(s)
University of Chicago, Chicago, USA. University of Texas, Austin, USA. exas A&M University—Qatar, Doha, Qatar.
University of Chicago, Chicago, USA. University of Texas, Austin, USA. exas A&M University—Qatar, Doha, Qatar.
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.
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.
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.
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.
References
[1] O. Perron, “The Theory of Matrices,” Mathematical Annalem, Vol. 64, No. 2, 1907, pp. 248-263. [2] G. Frobenius, “About Arrays of Non-negative Elements,” Reimer, Berlin, 1912. [3] S. U. Pillai, T. Suel and S. Cha, “The Perron-Frobenius Theorem: Some of Its Applications,” IEEE in Signal Processing Magazine, Vol. 22, No. 2, 2005, pp. 62-75. [4] J. Rohn, “Explicit Inverse of an Interval Matrix with Unit Midpoint,” Electronic Journal of Linear Algebra, Vol. 22, 2011, pp. 138-150. [5] J. Rohn, “A Handbook of Results on Interval Linear Problems,” 2005. http://uivtx.cs.cas.cz/ rohn/publist/!handbook.pdf [6] F. R. Gantmache, “The Theory of Matrices, Volume 2,” AMS Chelsea Publishing, Providence, 2000. [7] University of Nebraska-Lincoln, “Proof of Perron-Frobenius Theorem,” 2008. http://www.math.unl.edu/~bdeng1/Teaching/math428/Lecture%20Notes/PFTheorem.pdf [8] A. Borobia and U. R. Trfas, “A Geometric Proof of the Perron-Frobenius Theorem,” Revista Matematica de la, Vol. 5, No. 1, 1992, pp. 57-63. [9] H. Samelson, “On the Perron-Frobenius Theorem,” The Michigan Mathematical Journal, Vol. 4, No. 1, 1957, pp. 57-59. [10] T. Zahng, K. H. Law and G. H. Golub, “On the Homotopy Method for Symmetric Modified Generalized Eigenvalue Problems,” 1996. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.49.7261 [11] M. T. Chu, “A Note on the Homotopy Method for Linear Algebraic Eigenvalue Problems,” North Carolina State University, Raleigh, 1987. [12] P. Brockman, T. Carson, Y. Cheng, T. M. Elgindi, K. Jensen, X. Zhoun and M. B. M. Elgindi, “Homotopy Method for the Eigenvalues of Symmetric Tridiagonal Matrices,” Journal of Computational and Applied Mathematics, Vol. 237, No. 1, 2012, pp. 644-653. doi:10.1016/j.cam.2012.08.010
[1] O. Perron, “The Theory of Matrices,” Mathematical Annalem, Vol. 64, No. 2, 1907, pp. 248-263. [2] G. Frobenius, “About Arrays of Non-negative Elements,” Reimer, Berlin, 1912. [3] S. U. Pillai, T. Suel and S. Cha, “The Perron-Frobenius Theorem: Some of Its Applications,” IEEE in Signal Processing Magazine, Vol. 22, No. 2, 2005, pp. 62-75. [4] J. Rohn, “Explicit Inverse of an Interval Matrix with Unit Midpoint,” Electronic Journal of Linear Algebra, Vol. 22, 2011, pp. 138-150. [5] J. Rohn, “A Handbook of Results on Interval Linear Problems,” 2005. http://uivtx.cs.cas.cz/ rohn/publist/!handbook.pdf [6] F. R. Gantmache, “The Theory of Matrices, Volume 2,” AMS Chelsea Publishing, Providence, 2000. [7] University of Nebraska-Lincoln, “Proof of Perron-Frobenius Theorem,” 2008. http://www.math.unl.edu/~bdeng1/Teaching/math428/Lecture%20Notes/PFTheorem.pdf [8] A. Borobia and U. R. Trfas, “A Geometric Proof of the Perron-Frobenius Theorem,” Revista Matematica de la, Vol. 5, No. 1, 1992, pp. 57-63. [9] H. Samelson, “On the Perron-Frobenius Theorem,” The Michigan Mathematical Journal, Vol. 4, No. 1, 1957, pp. 57-59. [10] T. Zahng, K. H. Law and G. H. Golub, “On the Homotopy Method for Symmetric Modified Generalized Eigenvalue Problems,” 1996. http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.49.7261 [11] M. T. Chu, “A Note on the Homotopy Method for Linear Algebraic Eigenvalue Problems,” North Carolina State University, Raleigh, 1987. [12] P. Brockman, T. Carson, Y. Cheng, T. M. Elgindi, K. Jensen, X. Zhoun and M. B. M. Elgindi, “Homotopy Method for the Eigenvalues of Symmetric Tridiagonal Matrices,” Journal of Computational and Applied Mathematics, Vol. 237, No. 1, 2012, pp. 644-653. doi:10.1016/j.cam.2012.08.010