Degree Complexity of Matrix Inversion

TL;DR

This paper investigates the degree growth rate of iterates of a birational map formed by composing the matrix inverse and Hadamard inverse, providing insights into the algebraic complexity of these transformations.

Contribution

It determines the degree complexity of the map K, formed by composing matrix inverse and Hadamard inverse, revealing its exponential growth rate.

Findings

Degree growth of iterates is exponential.

Explicit calculation of the degree complexity.

Insights into algebraic dynamics of matrix transformations.

Abstract

For a q by q matrix x=(x_{i,j}) we let J(x)=(x_{i,j}^{-1}) be the Hadamard inverse, which takes the reciprocal of the elements of x . We let I(x)=(x_{i,j})^{-1} denote the matrix inverse, and we define K=I\circ J to be the birational map obtained from the composition of these two involutions. We consider the iterates K^n=K\circ...\circ K and determine degree complexity of K, which is the exponential rate of degree growth of the degrees of the iterates.