In this paper we study the positive definiteness of meet and join matrices using a novel approach. When the set S n is meet closed, we give a sufficient and necessary condition for the positive definiteness of the matrix (S n ) f . From this condition we obtain some sufficient conditions for positive definiteness as corollaries. We also use graph theory and show that by making some graph theoretic assumptions on the set S n we are able to reduce the assumptions on the function f while still preserving the positive definiteness of the matrix (S n ) f . Dual theorems of these results for join matrices are also presented. As examples we consider the so-called power GCD and power LCM matrices as well as MIN and MAX matrices. Finally we give bounds for the eigenvalues of meet and join matrices in cases when the function f possesses certain monotonic behaviour.
Let T = {z 1 , z 2 , . . . , zn} be a finite multiset of real numbers, where z 1 ≤ z 2 ≤ · · · ≤ zn. The purpose of this article is to study the different properties of MIN and MAX matrices of the set T with min(z i , z j ) and max(z i , z j ) as their ij entries, respectively. We are going to do this by interpreting these matrices as so-called meet and join matrices and by applying some known results for meet and join matrices. Once the theorems are found with the aid of advanced methods, we also consider whether it would be possible to prove these same results by using elementary matrix methods only. In many cases the answer is positive.
We give a set of axioms to establish a perpendicularity relation in an Abelian group and then study the existence of perpendicularities in(ℤn,+)and(ℚ+,·)and in certain other groups. Our approach provides a justification for the use of the symbol⊥denoting relative primeness in number theory and extends the domain of this convention to some degree. Related to that, we also consider parallelism from an axiomatic perspective.
Let (P, ) be a lattice and f a complex-valued function on P . We define meet and join matrices on two arbitrary subsets X and Y of PHere we present expressions for the determinant and the inverse of [X, Y ] f . Our main goal is to cover the case when f is not semimultiplicative since the formulas presented earlier for [X, Y ] f cannot be applied in this situation. In cases when f is semimultiplicative we obtain several new and known formulas for the determinant and inverse of (X, Y ) f and the usual meet and join matrices (S) f and [S] f . We also apply these formulas to LCM, MAX, GCD and MIN matrices, which are special cases of join and meet matrices.
In this article we give bounds for the eigenvalues of a matrix, which can be
seen as a common generalization of meet and join matrices and therefore also as
a generalization of both GCD and LCM matrices. Although there are some results
concerning the factorizations, the determinant and the inverse of this
so-called combined meet and join matrix, the eigenvalues of this matrix have
not been studied earlier. Finally we also give a nontrivial lower bound for a
certain constant $c_n$, which is needed in calculating the above-mentioned
eigenvalue bounds in practice. So far there are no such lower bounds to be
found in the literature
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