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On GCD, LCM and Hilbert matrices and their applications. (English) Zbl 1072.15026

Recall that the Hilbert matrix is the Hadamard inverse of the Hankel matrix. The authors give different inequalities between norms of the GCD, LCM and Hilbert matrices. The results depend upon the functions related to the number theory. For example, if \(G\) is an \(n\times n\) GCD matrix, then \(\det(G)= \varphi(1) \varphi(2) \cdots \varphi(n)\), \(\varphi\) being Euler’s totient function, and moreover \(\det(G)^{1/n}\leq {1\over\sqrt{n}}\| G\|_F\), where \(\| G\|_F\) denotes the Frobenius norm of \(G\), i.e. the norm of \(G\) as the vector of \(n^2\) Euclidean space.

MSC:

15A45 Miscellaneous inequalities involving matrices
15B57 Hermitian, skew-Hermitian, and related matrices
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors
11C20 Matrices, determinants in number theory
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References:

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