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

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

2003-03-05

arxiv.org/abs/nlin/0303007v2

Nonlinear Sciences

Exactly Solvable and Integrable Systems

10 pages; a few small improvements in this revised version

Scientific paper

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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