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

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.

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I. G. Korepanov. 2003-03-13. Exact solution for a matrix dynamical system with usual and Hadamard inverses. https://arxiv.org/abs/nlin/0303007

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