Solak, Süleyman; Türkmen, Ramazan; Bozkurt, Durmuş On GCD, LCM and Hilbert matrices and their applications. (English) Zbl 1072.15026 Appl. Math. Comput. 146, No. 2-3, 595-600 (2003). 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. Reviewer: Witold Wiȩsław (Wrocław) Cited in 2 Documents 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 Keywords:Norm; Hilbert matrix; GCD matrix; LCM matrix; Hadamard inverse; Hankel matrix; inequalities; greatest common divisor; lowest common multiplier; Euler’s totient function PDFBibTeX XMLCite \textit{S. Solak} et al., Appl. Math. Comput. 146, No. 2--3, 595--600 (2003; Zbl 1072.15026) Full Text: DOI References: [1] E. Altınışık, The characterization of the almost Hilbert-Smith matrices, Ph.D. Thesis, Konya, Turkey, 2001; E. Altınışık, The characterization of the almost Hilbert-Smith matrices, Ph.D. Thesis, Konya, Turkey, 2001 [2] Reams, R., Hadamard inverses, square roots and product of almost semidefinite matrices, Linear Algebra and Its Applications, 288, 35-43 (2000) · Zbl 0933.15006 [3] Haukkanen, P.; Wang, J.; Sillanpaa, J., On Smith’s Determinant, Linear Algebra and Its Applications, 258, 169-251 (1997) · Zbl 0883.15002 [4] Horn, R. A.; Johnson, C. R., Topics in Matrix Analysis (1991), Cambridge University Press · Zbl 0729.15001 [5] Mathias, R., The Spectral Norm of a Nonnegative Matrix, Linear Algebra and Its Applications, 131, 269-284 (1990) · Zbl 0705.15012 [6] D. Taşçı, Relations between norms and determinants, Science of Journal 12 Konya, Turkey, (1994); D. Taşçı, Relations between norms and determinants, Science of Journal 12 Konya, Turkey, (1994) · Zbl 0865.15024 [7] R. Moenck, On computing closed forms for summations, in: Proceedings of the 1977 MACSYMA Users’ Conference, NASA CP-2012, 1977 (paper no. 23 of this compilation); R. Moenck, On computing closed forms for summations, in: Proceedings of the 1977 MACSYMA Users’ Conference, NASA CP-2012, 1977 (paper no. 23 of this compilation) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.