Dynamics of products of nonnegative matrices
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Abstract
The aim of this manuscript is to understand the dynamics of products of nonnegative matrices. We extend a well known consequence of the Perron-Frobenius theorem on the periodic points of a nonnegative matrix to products of finitely many nonnegative matrices associated to a word and later to products of nonnegative matrices associated to a word, possibly of infinite length.
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How to Cite
Jayaraman, S., Prajapaty, Y., & Sridharan, S. (2022). Dynamics of products of nonnegative matrices. Extracta Mathematicae, 37(2), 223-242. https://doi.org/10.17398/2605-5686.37.2.223
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Section
Dynamical Systems
References
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[2] M. Akian, S. Gaubert, B. Lemmens, Stability and convergence in discrete convex monotone dynamical systems, J. Fixed Point Theory Appl. 9 (2011), 295 – 325.
[3] J. Bernik, R. Drnovsek, T. Kosir, T. Laffey, G. MacDonald, R. Meshulam, M. Omladic, H. Radjavi, Common fixed points and common eigenvectors for sets of matrices, Linear Multilinear Algebra 53 (2005), 137 – 146.
[4] M.P. Drazin, J.W. Dungey, K.W. Grunberg, Some theorems on commutative matrices, J. London Math. Soc. 26 (1951), 221 – 228.
[5] R.A. Horn, C.R. Johnson, “ Matrix Analysis ”, Second edition, Cambridge University Press, Cambridge, 2013.
[6] B.P. Kitchens, “ Symbolic Dynamics: One-sided, Two-sided and Countable State Markov Shifts ”, Universitext, Springer-Verlag, Berlin, 1998.
[7] B. Lemmens, Nonlinear Perron-Frobenius theory and dynamics of cone maps, in “ Positive Systems ”, Lect. Notes Control Inf. Sci., 341, Springer, Berlin, 2006, 399 – 406.
[8] B. Lemmens, R.D. Nussbaum, “ Nonlinear Perron-Frobenius Theory ”, Cambridge Tracts in Mathematics, 189, Cambridge University Press, Cambridge, 2012.
[9] R.D. Nussbaum, S.M. Verduyn Lunel, Generalizations of the Perron-Frobenius theorem for nonlinear maps, Mem. Amer. Math. Soc. 138 (1999), no. 659, viii+98.
[10] D. Shemesh, Common eigenvectors of two matrices, Linear Algebra Appl. 62 (1984), 11 – 18.
[11] X. Wang, Z. Cheng, Infinite products of uniformly paracontracting matrices, Linear Multilinear Algebra 64 (2016), 856 – 862.
[12] X. Zhan, “ Matrix Theory ”, Graduate Studies in Mathematics, 147, American Mathematical Society, Providence, RI, 2013.