Generating Orthogonal Matrices with Rational Elements

Title: Generating Orthogonal Matrices with Rational Elements

Research Question: How can we generate orthogonal matrices with rational elements using an algorithm?

Methodology: The research uses the concept of stereographic projection, which is a technique used in geometry to map points on a sphere to points on a plane. This method is applied to generate orthogonal matrices with rational elements.

Results: The study proves a theorem describing the structure of an arbitrary matrix from the group SO(n,Q), which is a subgroup of the orthogonal group O(n,R) consisting of orthogonal matrices with rational elements. This theorem yields an algorithm for generating such matrices using random number routines.

Implications: This research has implications in various fields such as computer science, mathematics, and physics. It provides a method for generating orthogonal matrices with rational elements, which can be used in algorithms and calculations that require such matrices. Additionally, the study contributes to the understanding of the structure of orthogonal matrices with rational elements.

In conclusion, this research presents an algorithm for generating orthogonal matrices with rational elements using the concept of stereographic projection. This algorithm can be used in various fields that require such matrices, and it contributes to the understanding of the structure of orthogonal matrices with rational elements.

Link to Article: https://arxiv.org/abs/0201007v2 Authors: arXiv ID: 0201007v2