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

TL;DR

This paper derives explicit solutions for a matrix dynamical system involving usual and Hadamard inverses, providing full solutions for 3x3 matrices and partial solutions for higher dimensions.

Contribution

It offers the first explicit solutions for the system generated by the product of matrix and Hadamard inversions, including a theta-function based ansatz for larger matrices.

Findings

Explicit solution for n=3 matrices.

Theta-function ansatz for n=4 matrices.

Partial solutions for higher n.

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.