GCD matrix

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

GCD matrix example

Let $S=\{x_{1},x_{2},\ldots ,x_{n}\}$ be a set 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 $[S]$ is defined analogously.

The study of GCD type matrices originates from HJS Smith (1875) 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.

Conjectures

Bourque-Ligh Conjecture: Bourque and Ligh (1992) conjectured that the LCM matrix on a GCD-closed set S is nonsingular. This conjecture was shown to be false by Haukkanen, Wang and Sillanpää (1997) and subsequently by Hong (1999). A lattice-theoretic approach is provided by Korkee, Mattila and Haukkanen (2019).

References

K. Bourque; S. Ligh (1992). "On GCD and LCM matrices". Linear Algebra and Its Applications. 174: 65–74.

P. Haukkanen; J. Wang; J. Sillanpää (1997). "On Smith's determinant". Linear Algebra and Its Applications. 258: 251–269.

S. Hong (1999). "On the Bourque–Ligh conjecture of least common multiple matrices". Journal of Algebra. 218: 216–228.

I. Korkee; M. Mattila; P. Haukkanen (2019). "A lattice-theoretic approach to the Bourque–Ligh conjecture". Linear and Multilinear Algebra. 67: 2471–2487.

H. J. S. Smith (1875). "On the value of a certain arithmetical determinant". Proceedings of the London Mathematical Society. 1: 208–213.