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On Bounds for the Smallest and the Largest Eigenvalues of GCD and LCM Matrices

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In this paper, improving a famous result of Wolkowicz and Styan for the GCD matrix (Sn)(S_n) and the LCM matrix [Sn][S_n] defined on Sn={1,2,,n}S_n=\{1,2,\ldots,n\}, we present new upper and lower bounds for the smallest and the largest eigenvalues of (Sn)(S_n) and [Sn][S_n] in terms of particular arithmetical functions.
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... In this paper, introducing a new constant cn(a), we have expanded the results on the nding all minimizing matrices of the constant cn and its asymptotic behaviour to a larger class of matrices. We do not reckon that our constant cn(a) could be used in eigenvalue estimation of GCD and related matrices as cn was used in the literature, see [3,6,8,9,11,12]. However, it seems possible that the techniques of this paper could be applied to a larger class of matrices than those considered in this present paper. ...
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... Namely, if ∀a ∈ S and d|a then d ∈ S . The matrix ⟨S⟩ g has been defined as ⟨S⟩ g = (s ij ) n×n ,s ij = g((a i , a j )) (i, j = 1, 2, … , n) and g((a i , a j )) refers to the value of the arithmetic function g on the greatest common factor gcd(a i , a j ) [1,2,7,8,12,13,21]. The following conclusions are proved by H.J.S. Smith [7]: if the set S is hypothetical to FC, then completely satisfactory formula det ⟨S⟩ g = (g ⊙ )(a 1 )(g ⊙ )(a 2 ) ⋯ (g ⊙ )(a n ) holds, where g ⊙ just happens to be the convolution between g and the mobius function . ...
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