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Nonlinear Sciences > Exactly Solvable and Integrable Systems

arXiv:nlin/0303007 (nlin)
[Submitted on 5 Mar 2003 (v1), last revised 13 Mar 2003 (this version, v2)]

Title:Exact solution for a matrix dynamical system with usual and Hadamard inverses

Authors:I.G. Korepanov
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Abstract: Let A be an n*n matrix with entries a_ij in the field C. Consider the following two involutive operations on such matrices: the matrix inversion I: A -> A^-1 and the element-by-element (or Hadamard) inversion J: a_ij -> a_ij^-1. We study the algebraic dynamical system generated by iterations of the product JI. In the case n=3, we give the full explicit solution for this system in terms of the initial matrix A. In the case n=4, we provide an explicit ansatz in terms of theta-functions which is full in the sense that it works for a Zariski open set of initial matrices. This ansatz also generalizes for higher n where it gives partial solutions.
Comments: 10 pages; a few small improvements in this revised version
Subjects: Exactly Solvable and Integrable Systems (nlin.SI)
Cite as: arXiv:nlin/0303007 [nlin.SI]
  (or arXiv:nlin/0303007v2 [nlin.SI] for this version)
  https://doi.org/10.48550/arXiv.nlin/0303007

Submission history

From: Igor G. Korepanov [view email]
[v1] Wed, 5 Mar 2003 09:45:55 UTC (9 KB)
[v2] Thu, 13 Mar 2003 09:29:43 UTC (9 KB)
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