Core Concepts

The paper proves that there are only finitely many Erdős matrices in any dimension and that these matrices can have only rational entries, resolving a question posed by Erdős and a recent conjecture.

Abstract

**Bibliographic Information:**Tripathi, R. (2024). Some observations on Erdős matrices. arXiv:2410.06612v1 [math.MG].**Research Objective:**This paper aims to address two open questions regarding Erdős matrices: (1) Are there finitely many Erdős matrices in any given dimension? (2) Do Erdős matrices have only rational entries?**Methodology:**The paper utilizes properties of bistochastic matrices, permutation matrices, the Frobenius norm, and concepts from convex geometry, particularly affine independence, to analyze and characterize Erdős matrices.**Key Findings:**The author proves that for every dimension 'n', there exist only a finite number of Erdős matrices. The paper further demonstrates that all Erdős matrices can be expressed as convex combinations of linearly independent permutation matrices, leading to the conclusion that Erdős matrices can only possess rational entries.**Main Conclusions:**The paper definitively resolves the two open questions regarding Erdős matrices. The finiteness of Erdős matrices in any given dimension is established, and it is shown that these matrices can only have rational entries.**Significance:**This paper significantly contributes to the understanding of Erdős matrices, a topic within the study of bistochastic matrices, which hold relevance in various fields like probability theory and graph theory. The findings provide a crucial theoretical basis for further research in this area.**Limitations and Future Research:**While the paper successfully proves the finiteness of Erdős matrices, it provides an upper bound for their quantity that is acknowledged to be far from sharp. Further research could focus on refining this bound and exploring the asymptotic behavior of the number of Erdős matrices as the dimension increases.

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Stats

For each dimension 'n', the maximum of Δn (a function related to the Frobenius norm and maximal trace of a bistochastic matrix) on the set of n x n bistochastic matrices is (n-1)/4.
Every n x n bistochastic matrix can be represented as a convex combination of at most (n-1)^2 + 1 permutation matrices.

Quotes

"We prove that for every n, there are only finitely many n × n Erdős matrices."
"We also prove that Erdős matrices can have only rational entries. This answers a question of [BMM24]."

Key Insights Distilled From

by Raghavendra ... at **arxiv.org** 10-10-2024

Deeper Inquiries

It's certainly possible! Here's a breakdown of why and how:
Core Techniques and Their Potential:
The paper leverages several core ideas that extend beyond bistochastic matrices:
Convexity: The use of convex combinations and Carathéodory's theorem is fundamental. Many matrix classes have convexity properties (e.g., positive semidefinite matrices, stochastic matrices).
Affine Independence: This concept is crucial for establishing the uniqueness of representations and solutions. It applies to vectors in general vector spaces, not just matrices.
Inner Products and Norms: The Frobenius norm and its connection to the trace are exploited. These concepts are standard in matrix analysis and have analogs in other settings.
Potential Applications:
Here are some matrix classes where similar techniques might be fruitful:
Stochastic Matrices: These generalize bistochastic matrices (columns sum to 1, but not necessarily rows). Analyzing "Erdős-like" properties in this broader class could be interesting.
Orthostochastic Matrices: Obtained by taking entry-wise squares of orthogonal matrices. The connection to permutation matrices might be less direct, but convexity arguments could still apply.
Doubly Substochastic Matrices: Entries are non-negative, and row and column sums are at most 1. The relaxation of the sum constraint might lead to different but related results.
Challenges and Adaptations:
Structure: The specific structure of bistochastic matrices (non-negative, fixed sums) is heavily used. Adaptations would be needed for other classes.
Permutation Matrix Analog: The role of permutation matrices as extreme points is key. Finding analogous "building blocks" for other matrix classes is essential.

Absolutely! The intimate link between Erdős matrices and permutation matrices strongly suggests deep connections to combinatorics. Here are some potential avenues for exploration:
1. Permutation Patterns and Matrix Structure:
Question: Can we classify Erdős matrices based on forbidden patterns in their associated permutation matrices? For example, do certain permutation patterns prevent a matrix from being an Erdős matrix?
Connection: This relates to the study of permutation patterns in combinatorics, which has rich connections to other areas like graph theory and sorting algorithms.
2. Graph Representations:
Idea: Represent permutation matrices as bipartite graphs (rows and columns as vertices, a 1 entry corresponds to an edge).
Question: Do the graphs corresponding to Erdős matrices exhibit special properties? For instance, do they have specific subgraph restrictions or high degrees of symmetry?
3. Combinatorial Optimization:
Observation: The definition of Erdős matrices involves maximizing a sum over permutations, reminiscent of assignment problems or matching problems in graphs.
Question: Can we formulate the search for Erdős matrices as an optimization problem on a suitable combinatorial structure? This could lead to new algorithms or insights into their complexity.
4. Designs and Latin Squares:
Analogy: Bistochastic matrices are related to the concept of Latin squares, which are square grids filled with symbols such that each symbol appears exactly once in each row and column.
Question: Do Erdős matrices correspond to Latin squares with special properties? Could they be used to construct new combinatorial designs?

Relaxing the non-negativity constraint significantly changes the problem and likely invalidates the finiteness and rationality results. Here's why:
1. Loss of Convexity:
Key Point: The set of matrices with entries summing to 1 in each row and column, but without the non-negativity constraint, is no longer a convex set.
Consequence: Carathéodory's theorem, which was crucial for representing matrices as convex combinations of permutation matrices, no longer applies. We lose a fundamental tool.
2. Infinite Solutions:
Example: Consider the 2x2 case. Without non-negativity, the equation for an Erdős matrix becomes:
(a² + b² + c² + d²) = max(a + d, b + c), where a + b = c + d = 1
Observation: We can find infinitely many solutions to this equation, even irrational ones. For instance, we can freely choose a and d within certain bounds, and the other entries are determined.
3. Rationality Unlikely:
Reasoning: The rationality result heavily relied on the fact that the matrix M (with inner products of permutation matrices) had integer entries. Without non-negativity, the entries of M can be arbitrary real numbers, making it unlikely that the solution x will always be rational.
In summary, relaxing the non-negativity constraint fundamentally alters the problem. We lose the guarantees of finiteness and rationality, and the combinatorial connections become less clear.

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