The Sinkhorn Limit of Positive 3 × 3 Matrices
ISEF · 2023 Mathematics
Overview
A positive square matrix is an n × n array of positive real number entries. The Sinkhorn-Knopp alternate minimization algorithm transforms these matrices into their respective doubly stochastic Sinkhorn limits. This effect is analyzed using both numerical and symbolic data. A modified version of Nathanson’s explicit formula for the Sinkhorn limit of general positive 2 × 2 matrices is introduced. Then a methodology inspired by Nathanson and Zeilberger that utilizes Buchberger’s algorithm to compute Gröbner bases in order to find an explicit formula for the Sinkhorn limit of general positive 3 × 3 matrices is discussed. After implementing this methodology, a desired explicit formula is provided and analyzed. Focal parts of this formula are symbolically analyzed for applications in finding explicit formulae for the Sinkhorn limit of n × n matrices where n>3.
Competition history
- ISEF 2023
Resources
Related projects
ISEF · 2026
Eigenvalue Distributions of Stochastic Matrices and the Karpelevic Region
CWSF · 2026
A Novel Approach to Sinkhorn ε-Annealing: A 3.62x Speedup for the Optimal Transport Problem
ISEF · 2026
Universal Matrices for Counting Fibo-Multinomial and C-Multinomial Coefficients With a Cryptographic Application
ISEF · 2023
New P Formula To Calculate the Determinants of 3x3 Matrices in a More Algebraic Way
Closest projects by meaning, across every fair and year in the corpus.
Source: Regeneron International Science and Engineering Fair