[Paper Review] Asymptotic behavior of the smallest eigenvalue of matrices associated with completely even functions (mod r)

Shaofang Hong, Raphael Loewy|arXiv (Cornell University)|Aug 29, 2008

Matrix Theory and Algorithms26 references4 citations

TL;DR

This paper investigates the asymptotic behavior of the smallest eigenvalue of matrices associated with completely even functions modulo r, focusing on Dirichlet convolution and tensor product structures. It establishes that for certain classes of multiplicative functions and sequences containing arithmetic progressions, the smallest eigenvalue tends to zero as n increases, with sharp bounds derived via Cauchy's interlacing inequalities and Mertens-type estimates.

ABSTRACT

In this paper we present systematically analysis on the smallest eigenvalue of matrices associated with completely even functions (mod $r$). We obtain several theorems on the asymptotic behavior of the smallest eigenvalue of matrices associated with completely even functions (mod $r$). In particular, we get information on the asymptotic behavior of the smallest eigenvalue of the famous Smith matrices. Finally some examples are given to demonstrate the main results.

Motivation & Objective

• To analyze the asymptotic behavior of the smallest eigenvalue of matrices derived from completely even functions modulo r.
• To extend prior results on Smith matrices and reciprocal power LCM matrices to broader classes of arithmetical functions.
• To characterize conditions under which the smallest eigenvalue tends to zero as matrix size n → ∞.
• To establish sharp bounds using spectral theory and number-theoretic tools like Dirichlet convolution and Mertens’ theorem.

Proposed method

• Utilizes the framework of completely even functions modulo r, defined via gcd-based evaluation f((m,r),r).
• Applies Dirichlet convolution and Möbius inversion to transform matrix entries into arithmetic functions with positive convolution outputs.
• Employs tensor product decomposition to analyze matrix structure and eigenvalue distribution.
• Applies Cauchy’s interlacing inequalities to relate eigenvalues across nested matrices and infer convergence behavior.
• Uses Mertens’ theorem and Dirichlet’s theorem on primes in arithmetic progressions to estimate sums over multiplicative functions.
• Analyzes eigenvalue decay by linking the growth of f(x_i) to convergence of ∑1/f(x_i), particularly when ∑1/f(x_i) diverges.

Experimental results

Research questions

• RQ1 Under what conditions on the sequence {x_i} and the function f does the smallest eigenvalue of the matrix ((f^{(c)}*μ^{(d)})(x_i,x_j)) tend to zero as n → ∞?
• RQ2 How does the asymptotic behavior of the smallest eigenvalue depend on the multiplicative and increasing properties of f on subsequences with bounded gcd?
• RQ3 What role does the divergence or convergence of ∑1/f(x_i) play in determining the positivity or vanishing of the smallest eigenvalue in the limit?
• RQ4 Can sharp lower bounds be established for the smallest eigenvalue when f grows faster than x^ε for ε > 1?
• RQ5 Is there a characterization of sequences {x_i} such that the smallest eigenvalue remains bounded away from zero when f is multiplicative and ∑1/f(x_i) < ∞?

Key findings

• For any strictly increasing sequence {x_i} containing an arithmetic progression {a+bi} as a subsequence, and for 0 ≤ ε ≤ 1, the q-th smallest eigenvalue of the matrix (i,j)^{2ε}/(i^ε j^ε) tends to zero as n → ∞.
• If f is multiplicative, increasing on a subsequence with pairwise gcd equal to the first term, and ∑1/f(x_i') = ∞, then the smallest eigenvalue λ_n^{(1)}(1,0) → 0 as n → ∞.
• When f satisfies f(x_i) ≥ C x_i^ε for ε > 1 and C > 0, the paper conjectures that λ_n^{(1)}(c,d) remains bounded away from zero, suggesting a threshold behavior at ε = 1.
• For sequences with ∑1/f(x_i) < ∞, the paper conjectures that λ_n^{(1)}(c,d) > 0 in the limit, implying eigenvalue stability under summability conditions.
• The smallest eigenvalue of the power GCD matrix ((x_i,x_j)^ε) tends to zero if ∑1/x_i^ε = ∞, and the paper conjectures it remains positive if the sum converges.
• The results are sharp in the sense that eigenvalue decay is governed by the divergence of ∑1/f(x_i), with Cauchy’s interlacing inequalities providing the key technical bridge to eigenvalue monotonicity.