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 ...

, a greatest common divisor matrix (sometimes abbreviated as GCD matrix) is a

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...

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

\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 In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...

Bourque–Ligh conjecture

conjectured that the LCM matrix on a GCD-closed set

S

is nonsingular. This conjecture was shown to be false by and subsequently by . A lattice-theoretic approach is provided by .

References

{{reflist, refs = {{cite journal , last1 = Bourque , first1 = K. , last2 = Ligh , first2 = S. , title = On GCD and LCM matrices , journal = Linear Algebra and Its Applications , year = 1992 , volume = 174 , pages = 65–74 , doi = 10.1016/0024-3795(92)90042-9 , doi-access = free {{cite journal , last1 = Haukkanen , first1 = P. , last2 = Wang , first2 = J. , last3 = Sillanpää , first3 = J. , title = On Smith's determinant , journal = Linear Algebra and Its Applications , year = 1997 , volume = 258 , pages = 251–269 , doi = 10.1016/S0024-3795(96)00192-9 , doi-access = free {{cite journal , last = Hong , first = S. , title = On the Bourque–Ligh conjecture of least common multiple matrices , journal = Journal of Algebra , year = 1999 , volume = 218 , pages = 216–228 , doi = 10.1006/jabr.1998.7844 , doi-access = free {{cite journal , last1 = Korkee , first1 = I. , last2 = Mattila , first2 = M. , last3 = Haukkanen , first3 = P. , title = A lattice-theoretic approach to the Bourque–Ligh conjecture , journal = Linear and Multilinear Algebra , year = 2019 , volume = 67 , issue = 12 , pages = 2471–2487 , doi = 10.1080/03081087.2018.1494695 , s2cid = 117112282 , url = https://trepo.tuni.fi/handle/10024/117430 {{cite journal , last = Smith , first = H. J. S. , title = On the value of a certain arithmetical determinant , journal = Proceedings of the London Mathematical Society , year = 1875 , volume = 1 , pages = 208–213 , doi = 10.1112/plms/s1-7.1.208, url = https://zenodo.org/record/1709912 Matrix theory Number theory