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GCD Matrix
In mathematics, a greatest common divisor matrix (sometimes abbreviated as GCD matrix) is a matrix. Definition Let S=(x_1, x_2,\ldots, x_n) be a list of positive integers. Then the n\times n matrix (S) having the greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ... \gcd(x_i, x_j) as its ij entry is referred to as the GCD matrix on S.The LCM matrix /math> is defined analogously. The study of GCD type matrices originates from who evaluated the determinant of certain GCD and LCM matrices. Smith showed among others that the determinant of the n\times n matrix (\gcd(i,j)) is \phi(1)\phi(2)\cdots\phi(n), where \phi is Euler's totient function. Bourque–Ligh conjecture conjectured that the LCM matrix on a GCD-closed set S is nonsingular. This conjectu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
