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A proof of a conjecture on monotonic behavior of the smallest and the largest eigenvalues of a number theoretic matrix. (English) Zbl 1307.15016

Summary: We investigate the monotonic behavior of the smallest eigenvalue \(t_n\) and the largest eigenvalue \(T_n\) of the \(n \times n\) matrix \(E_n^T E_n\), where the \(ij\)-entry of \(E_n\) is \(1\) if \(j | i\) and \(0\) otherwise. We present a proof of the Mattila-Haukkanen conjecture which states that for every \(n \in \mathbb{Z}^+\), \(t_{n + 1} \leq t_n\) and \(T_n \leq T_{n + 1}\). Also, we prove that \(\lim_{n \to \infty} t_n = 0\) and \(\lim_{n \to \infty} T_n = \infty\) and we give a lower bound for \(t_n\).

MSC:

15A23 Factorization of matrices
15B36 Matrices of integers
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
11A25 Arithmetic functions; related numbers; inversion formulas
11C20 Matrices, determinants in number theory
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