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Exact solution for a matrix dynamical system with usual and Hadamard inverses | | I.G. Korepanov
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5 Mar 2003 | | Subject: | Exactly Solvable and Integrable Systems | nlin.SI | | 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. | | Source: | arXiv, nlin.SI/0303007 | | Services: | Forum | Review | PDF | Favorites | |
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