On a Generalization of the Reciprocal Lcm Matrix
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Tarih
2002
Yazarlar
Dergi Başlığı
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Cilt Başlığı
Yayıncı
Erişim Hakkı
info:eu-repo/semantics/openAccess
Özet
Let S = {x,,, be a set of distinct positive integers. The nxn matrix 1/[S]=(s), where s=1/[x,,x,], the reciprocal of the least common multiple of x, and x,, is called the reciprocal least common multiple (reciprocal LCM) matrix on S. In this paper, we present a generalization of the reciprocal LCM matrix on S, that is the matrix 1/[S'], the ij- entry of which is 1/[x,,x,], where r is a real number. We obtain a structure theorem for 1/[S] and the value of the determinant of 1/[S"]. We also prove that 1/[S'] is positive definite if r>0. Then we calculate the inverse of 1/[S'] on a factor closed set. Finally, we show that the matrix [S']=([x,,x,]) defined on S is the product of an integral matrix and the generalized reciprocal LCM matrix 1/[S'] = (1/[x,,x,]) if S is factor closed and r is a positive integer.
Açıklama
Anahtar Kelimeler
İstatistik ve Olasılık, Matematik, The GCD matrix, the LCM matrix, the reciprocal GCD matrix, the reciprocal LCM matrix, Euler's totient function, Jordan's totient function, factor closed set
Kaynak
Communications Series A1: Mathematics and Statistics
WoS Q Değeri
Scopus Q Değeri
Cilt
51
Sayı
2
Künye
Altınışık, E., Taşcı, D., (2002). On a Generalization of the Reciprocal Lcm Matrix. Communications Series A1: Mathematics and Statistics, 51(2), 37-46.
