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Divisibility properties of power GCD matrices and power LCM matrices. (English) Zbl 1137.11017

Let \(S=\{x_1,x_2,\ldots,x_n\}\) be a set of \(n\) distinct positive integers, and let \(a\) and \(b\) be positive integers. The \(n\times n\) matrix \((S^a)\) having the \(a\)th power \((x_i,x_j)^a\) of the greatest common divisor of \(x_i\) and \(x_j\) as its \(ij\) entry is called the power GCD matrix on \(S\). The power LCM matrix \([S^a]\) is defined analogously. The author shows that if \(S\) is a divisor chain with \(n\geq 2\), then \((S^a)\) divides \((S^b)\) in the ring of \(n\times n\) matrices over the integers if and only if \(a\mid b\). Similar results for power LCM matrices and mixed cases are also obtained. The study of divisibility of GCD and related matrices was begun in K. Bourque and S. Ligh [Linear Algebra Appl. 174, 65–74 (1992; Zbl 0761.15013)]. For surveys of basic properties of GCD and related matrices see I. Korkee and P. Haukkanen [Linear Algebra Appl. 372, 127–153 (2003; Zbl 1036.06005)] and [J. Sándor and B. Crstici, Handbook of number theory II. Dordrecht: Kluwer Academic Publishers (2004; Zbl 1079.11001)].

MSC:

11C20 Matrices, determinants in number theory
15B36 Matrices of integers
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