In mathematics, a greatest common divisor matrix (sometimes abbreviated as GCD matrix) is a matrix.
1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 |
1 | 1 | 3 | 1 | 1 | 3 | 1 | 1 | 3 | 1 |
1 | 2 | 1 | 4 | 1 | 2 | 1 | 4 | 1 | 2 |
1 | 1 | 1 | 1 | 5 | 1 | 1 | 1 | 1 | 5 |
1 | 2 | 3 | 2 | 1 | 6 | 1 | 2 | 3 | 2 |
1 | 1 | 1 | 1 | 1 | 1 | 7 | 1 | 1 | 1 |
1 | 2 | 1 | 4 | 1 | 2 | 1 | 8 | 1 | 2 |
1 | 1 | 3 | 1 | 1 | 3 | 1 | 1 | 9 | 1 |
1 | 2 | 1 | 2 | 5 | 2 | 1 | 2 | 1 | 10 |
GCD matrix of (1,2,3,...,10) |
Let be a list of positive integers. Then the matrix having the greatest common divisor as its entry is referred to as the GCD matrix on .The LCM matrix is defined analogously.[1][2]
The study of GCD type matrices originates from Smith (1875) who evaluated the determinant of certain GCD and LCM matrices. Smith showed among others that the determinant of the matrix is , where is Euler's totient function.[3]
Bourque & Ligh (1992) conjectured that the LCM matrix on a GCD-closed set is nonsingular.[1] This conjecture was shown to be false by Haukkanen, Wang & Sillanpää (1997) and subsequently by Hong (1999).[4][2] A lattice-theoretic approach is provided by Korkee, Mattila & Haukkanen (2019).[5]