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On the Probability of Generating a Primitive matrix

Journal of Systems Science & Complexity 2024年第4期37卷 1755-1771页

作者： CHEN Jingwei FENG Yong LIU Yang WU Wenyuan Chongqing Key Laboratory of Automated Reasoning and Cognition Chongqing Institute of Green and Intelligent TechnologyChinese Academy of SciencesChongqing 400714China Chongqing College University of Chinese Academy of SciencesChongqing 400714China Key Laboratory of Advanced Manufacturing Technology of Ministry of Education Guizhou UniversityyGuiyang 550025China Information Science and Engineering Chongqing Jiaotong UniversityChongqing 400074China

Given a k×n integer primitive matrix A(i.e.,a matrix can be extended to an n×n unimodular matrix over the integers)with the maximal absolute value of entries‖A‖bounded by an integer λ from above,the authors study the probability that the m×n matrix extended from A by appending other m-k row vectors of dimension n with entries chosen randomly and independently from the uniform distribution over{0,1,…,λ-1}is still *** authors present a complete and rigorous proof of a lower bound on the probability,which is at least a constant for fixed m in the range[k+1,n-4].As an application,the authors prove that there exists a fast Las Vegas algorithm that completes a k×n primitive matrix A to an n×n unimodular matrix within expected O(n^(ω)log‖A‖)bit operations,where O is big-O but without log factors,ω is the exponent on the arithmetic operations of matrix multiplication.

关键词： integer matrix matrix completion probabilistic algorithm unimodular matrix

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Divisibility Properties of Power Matrices Associated with Arithmetic Functions on a Divisor Chain

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Algebra Colloquium 2022年第3期29卷 527-540页

作者： Long Chen Zongbing Lin Qianrong Tan School of Mathematics and Computer Science Panzhihua University PanzhihuaSichuan 617000China

Let a,b and n be positive integers withn≥2,f be an integer-valued arithmetic function,and the set S={x_(1),…,x_(n)}of n distinct positive integers be a divisor chain such that x_(1)|x_(2)|⋯|x_(n).We first show that the matrix(f_(a)(S))having f evaluated at the ath power(x_(i),x_(j))^(a) of the greatest common divisor of x_(i) and x_(j) as its i,j-entry divides the GCD matrix(f^(b)(S))in the ring M_(n)(Z)of n×n matrices over integers if and only if f^(b−a)(x_(1))∈Z and(f^(a)(x_(i))−f^(a)(x_(i−1)))divides(f^(b)(x_(i))−f^(b)(x_(i−1)))for any integer i with 2≤i≤***,we show that the matrix(f^(a)[S])having f evaluated at the ath power[x_(i),x_(j)]^(a) of the least common multiple of x_(i) and x_(j) as its i,j-entry divides the matrix(f^(b)[S])in the ring M_(n)(Z)if and only if f^(b−a)(x_(n))∈Z and(f^(a)(x_(i))−f^(a)(x_(i−1)))divides(f^(b)(x_(i))−f^(b)(x_(i−1)))for any integer i with2≤i≤***,we prove that the matrix(f^(a)(S))divides the matrix(f^(b)[S])in the ring M_(n)(Z)if and only if f^(a)(x_(1))|f^(b)(x_(i))and(f^(a)(x_(i))−f^(a)(x_(i−1)))|(f^(b)(x_(i))−f^(b)(x_(i−1)))for any integer i with 2≤i≤*** results extend and strengthen the theorems of Hong obtained in 2008.

关键词： divisor chain integer-valued arithmetic function integer matrix divisibility

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