Generating Orthogonal Matrices with Rational Elements: Difference between revisions

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Created page with "Title: Generating Orthogonal Matrices with Rational Elements Research Question: Can we develop an algorithm to generate orthogonal matrices with rational elements? Methodology: The researchers used the concept of stereographic projection, which is a technique used in geometry to map the surface of a sphere onto a flat surface. They applied this concept to the orthogonal matrices with rational elements. Results: The researchers proved that each orthogonal matrix O∈SO..."
 
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Title: Generating Orthogonal Matrices with Rational Elements
Title: Generating Orthogonal Matrices with Rational Elements


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


Methodology: The researchers used the concept of stereographic projection, which is a technique used in geometry to map the surface of a sphere onto a flat surface. They applied this concept to the orthogonal matrices with rational elements.
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 researchers proved that each orthogonal matrix O∈SO(n,R) can be represented as a product of n(n−1)/2 matrices of elementary rotations (1.1). They also developed an algorithm for constructing orthogonal matrices over the field of rational numbers, which means that each orthogonal matrix O∈SO(n,Q) could be obtained by applying this algorithm.
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 new method for generating orthogonal matrices with rational elements, which can be useful in algorithms that require such matrices. Additionally, the research has potential applications in quantum mechanics and general relativity.
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.


Link to Article: https://arxiv.org/abs/0201007v1
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:  
Authors:  
arXiv ID: 0201007v1
arXiv ID: 0201007v2


[[Category:Computer Science]]
[[Category:Computer Science]]
[[Category:Matrices]]
[[Category:Orthogonal]]
[[Category:Orthogonal]]
[[Category:Matrices]]
[[Category:Rational]]
[[Category:Rational]]
[[Category:Elements]]
[[Category:Elements]]
[[Category:N]]
[[Category:This]]

Latest revision as of 03:50, 24 December 2023

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

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