In mathematics, a greatest common divisor matrix (sometimes abbreviated as GCD matrix) is a matrix that may also be referred to as Smith's matrix. The study was initiated by H.J.S. Smith (1875). A new inspiration was begun from the paper of Bourque & Ligh (1992). This led to intensive investigations on singularity and divisibility of GCD type matrices. A brief review of papers on GCD type matrices before that time is presented in .
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.
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 matrix is , where is Euler's totient function.
conjectured that the LCM matrix on a GCD-closed set is nonsingular. This conjecture was shown to be false by and subsequently by . A lattice-theoretic approach is provided by .
The counterexample presented in is and that in is A counterexample consisting of odd numbers is . Its Hasse diagram is presented on the right below.
The cube-type structures of these sets with respect to the divisibility relation are explained in .
Let be a factor closed set. Then the GCD matrix divides the LCM matrix in the ring of matrices over the integers, that is there is an integral matrix such that , see . Since the matrices and are symmetric, we have . Thus, divisibility from the right coincides with that from the left. We may thus use the term divisibility.
There is in the literature a large number of generalizations and analogues of this basic divisibility result.
Some results on matrix norms of GCD type matrices are presented in the literature. Two basic results concern the asymptotic behaviour of the norm of the GCD and LCM matrix on .
Given , the norm of an matrix is defined as
Let . If , then
where
and for and . Further, if , then
where
Let be an arithmetical function, and let be a set of distinct positive integers. Then the matrix is referred to as the GCD matrix on associated with . The LCM matrix on associated with is defined analogously. One may also use the notations and .
Let be a GCD-closed set. Then
where is the matrix defined by
and is the diagonal matrix, whose diagonal elements are
Here is the Dirichlet convolution and is the Möbius function.
Further, if is a multiplicative function and always nonzero, then
where and are the diagonal matrices, whose diagonal elements are
and
If is factor-closed, then and .