Nonsingularity of the reciprocal power matrices on three quasi-coprime divisor chains
-
摘要: 首先给出定义在三个拟互素因子链上的倒数幂~GCD~矩阵和倒数幂~LCM 矩阵的行列式的计算公式, 由此证明定义在三个拟互素因子链~$S$~上且~$S$ 的最大公因子属于~$S$~时的倒数幂~GCD~矩阵和倒数幂~LCM~矩阵是非奇异的. 但当构成~$S$~的三个因子链不素时, 如此的结果不成立.
-
关键词:
- 三个拟互素因子链 /
- 最大型因子 /
- 倒数幂~GCD~矩阵 /
- 倒数幂~LCM~矩阵
Abstract: The authors first obtained formulae for the determinants of reciprocal power GCD matrix and reciprocal power LCM matrix defined on a set $S$ consisting of three quasi-coprime divisor chains with $\gcd (S)\in S$. Consequently, it was showed that the reciprocal power GCD matrix and the reciprocal power LCM matrix defined on $S$ with $\gcd (S)\in S$ are nonsingular. But such result is not true when $S$ consists of three divisor chains which are not quasi-coprime. -
[1] {1}BOURQUE K, LIGH S. On GCD and LCM matrices[J]. Linear Algebra Appl,1992, 174: 65-74.{2}CAO W. On Hong's conjecture for power LCM matrices[J]. CzechoslovakMath J, 2007, 57: 253-268.{3}HONG S. On the Bourque-Ligh conjecture of least common multiplematrices[J]. J Algebra, 1999, 218: 216-228.{4}HOGN S. Gcd-closed sets and determinants of matrices associated witharithmetical functions[J]. Acta Arith, 2002, 101: 321.{5}HOGN S. Nonsingularity of matrices associated with classes ofarithmetical functions[J]. J Algebra, 2004, 281: 1-14.{6}HOGN S. Divisibility properties of power GCD matrices and power LCMmatrices[J]. Linear Algebra Appl, 2008, 428: 1001-1008.{7}HOGN S, LEE Enoch K S. Asymptotic behavior of eigenvalues ofreciprocal power LCM matrices[J]. Glasgow Math J, 2008, 50(1):163-174.{8}HOGN S, LOEWY R. Asymptotic behavior of eigenvalues of greatestcommon divisor matrices[J]. Glasgow Math J, 2004, 46: 551-569.{9}HOGN S, LOEWY R. Asymptotic behavior of the smallest eigenvalue ofmatrices associated with completely even functions (mod r)[J].International J, Number Theory, 2011, 7: 1681-1704.{10}LIN Z, TAN Q. Determinants of Smith matrices on three coprimedivisor chains and divisibility[J]. Linear Multilinear Algebra,2012, 60: 475-486.{11}LIN Z, TAN Q. Nonsingularity of reciprocal power GCD matrices andreciprocal power LCM matrices on two quasi-coprime divisorchains[J]. J Sichuan Univ Nat Sci Ed, in press.{12}SMITH H J S. On the value of a certain arithmetical determinant[J].Proc London Math Soc, 1875-1876, 7: 208-212.{13}TAN Q. Divisibility among power GCD matrices and among power LCMmatrices on two coprime divisor chains[J]. Linear MultilinearAlgebra, 2010, 58: 659-671.{14}XU J, LI M. Divisibility among power GCD matrices and among powerLCM matrices on three coprime divisor chains[J]. Linear MultilinearAlgebra, 2011, 59: 773-788.
点击查看大图
计量
- 文章访问数: 1918
- HTML全文浏览量: 16
- PDF下载量: 1953
- 被引次数: 0