Article

On eigenvalues of meet and join matrices associated with incidence functions

Authors:
Pauliina Ilmonen at Aalto University
Pauliina Ilmonen
Pentti Haukkanen at Tampere University
Pentti Haukkanen
Jorma K Merikoski at Tampere University
Jorma K Merikoski
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Abstract

Let (P,⪯,∧) be a locally finite meet semilattice. Letbe a finite subset of P and let f be a complex-valued function on P. Then the n×n matrix (S)f, where((S)f)ij=f(xi∧xj),is called the meet matrix on S with respect to f. The join matrix on S with respect to f is defined dually on a locally finite join semilattice.In this paper, we give lower bounds for the smallest eigenvalues of certain positive definite meet matrices with respect to f on any set S. We also estimate eigenvalues of meet matrices respect to any f on meet closed set S and with respect to semimultiplicative f on join closed set S. The same is carried out dually for join matrices.

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... Bhatia [9] considered GCD matrices as an example of infinitely divisible matrices. Ilmonen, Haukkanen, and Merikoski [10] instigated the study eigenvalues of meet and join matrices associated with incidence functions. The eigenvalues of meet and join matrices have since been studied thoroughly in recent literature: Mattila and Haukkanen [11] studied positive definiteness and eigenvalues of meet and join matrices, and Mattila, Haukkanen, and Mäntysalo [12] considered the singularity of LCM-type matrices. ...
... To this end, we observe that the matrix E in (1) belongs to the matrix algebra K(n) of all n × n lower triangular 0, 1 matrices such that each main diagonal entry is equal to 1. Clearly every matrix X ∈ K(n) is real and nonsingular and thus XX T is positive definite. We now define the positive constants c n [8] and C n [10] depending only on n such that ...
... However, it has been shown recently that the representative matrices in K(n) corresponding to the constants c n and C n are known a priori. In the paper [10], it has been demonstrated that the symmetric n × n matrix M n satisfying ...
Generalized eigenvalue problems for meet and join matrices on semilattices
Preprint
  • May 2017
We study generalized eigenvalue problems for meet and join matrices with respect to incidence functions on semilattices. We provide new bounds for generalized eigenvalues of meet matrices with respect to join matrices under very general assumptions. The applied methodology is flexible, and it is shown in the case of GCD and LCM matrices that even sharper bounds can be obtained by applying the known properties of the divisor lattice. These results can also be easily modified for the dual problem of eigenvalues of join matrices with respect to meet matrices, which we briefly consider as well. We investigate the effectiveness of the obtained bounds for select examples involving number-theoretical lattices.
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... There are also various generalizations of these matrices, where the elements of S are taken from a suitably chosen partially ordered set, and gcd and lcm are replaced by meet and join. For references, see [1,4,5] and references therein. ...
... c n has been used to obtain various inequalities, cf. [1,4,5] and references therein. Our purpose here is to investigate it. ...
... Our purpose here is to investigate it. Ilmonen, Haukkanen and Merikoski [5] posed the following conjecture. ...
On the smallest singular value in the class of invertible lower triangular (0,1) matrices
Article
  • Aug 2020
  • LINEAR ALGEBRA APPL
Given an n×n real symmetric matrix A, let λn(A) denote its smallest eigenvalue. Let Kn denote the set of all n×n invertible, lower triangular (0,1) matrices, and cn:=min{λn(YYt):Y∈Kn}. Then, cn is the smallest singular value in Kn. Hong and Loewy introduced cn as a mean to obtain inequalities involving eigenvalues of certain GCD (greatest common divisor) and LCM (least common multiple) matrices. Since then, cn has been used in many papers to obtain additional spectral inequalities for GCD and LCM matrices, and their generalizations. Due to its wide spread, it became important to obtain good bounds for cn. In this paper we obtain such bounds, and consequently determine the asymptotic behavior of cn, proving a conjecture of Kaarnioja. Moreover, we prove the uniqueness of the matrix Y∈Kn for which cn is attained, proving a conjecture of Altinisik, Keskin, Yildiz and Demirbüken.
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... A closely related sequence was introduced by Ilmonen, Haukkanen, and Merikoski [3]: ...
... Example 1.1 (cf. [3]). Let (P, , ∧,0) be a locally finite meet semilattice, where is a partial ordering on the set P , ∧ denotes the meet (or greatest lower bound) of two elements in P , and0 ∈ P is the least element such that0 ...
... For example, in the case of the divisor lattice (Z + , |, gcd) and the identity function f (x) = x for x ∈ Z + , the function J P,f is precisely Euler's totient function. See [3] for a rigorous statement of this result and see [2] for the special case of greatest common divisor matrices. ...
Sharp bounds on the spectrum of triangular Boolean matrices
Preprint
  • Feb 2020
Let KnK_n be the set of all nonsingular n×nn\times n lower triangular Boolean (0,1) matrices. Hong and Loewy (2004) introduced the numbers cn=min{λ∣λ is an eigenvalue of XXT, X∈Kn},n∈Z+. c_n={\rm min}\{\lambda\mid \lambda~\text{is an eigenvalue of}~XX^{\rm T},~X\in K_n\},\quad n\in\mathbb{Z}_+. A related family of numbers was considered by Ilmonen, Haukkanen, and Merikoski (2008): Cn=max{λ∣λ is an eigenvalue of XXT, X∈Kn},n∈Z+. C_n={\rm max}\{\lambda\mid \lambda~\text{is an eigenvalue of}~XX^{\rm T},~X\in K_n\},\quad n\in\mathbb{Z}_+. These numbers can be used to bound the singular values of matrices belonging to KnK_n and they appear, e.g., in the estimation of the spectral radii of GCD and LCM matrices as well as their lattice-theoretic generalizations. In this paper, it is shown that for n odd, one has the lower bound cn≥1125φ−4n+225φ−2n−255nφ−2n−2325+n+225φ2n+255nφ2n+125φ4n, c_n\geq \frac{1}{\sqrt{\frac{1}{25}\varphi^{-4n}+\frac{2}{25}\varphi^{-2n}-\frac{2}{5\sqrt{5}}n\varphi^{-2n}-\frac{23}{25}+n+\frac{2}{25}\varphi^{2n}+\frac{2}{5\sqrt{5}}n\varphi^{2n}+\frac{1}{25}\varphi^{4n}}}, and for n even, one has cn≥1125φ−4n+425φ−2n−255nφ−2n−25+n+425φ2n+255nφ2n+125φ4n, c_n\geq \frac{1}{\sqrt{\frac{1}{25}\varphi^{-4n}+\frac{4}{25}\varphi^{-2n}-\frac{2}{5\sqrt{5}}n\varphi^{-2n}-\frac{2}{5}+n+\frac{4}{25}\varphi^{2n}+\frac{2}{5\sqrt{5}}n\varphi^{2n}+\frac{1}{25}\varphi^{4n}}}, where φ\varphi denotes the golden ratio. These lower bounds improve the estimates derived previously by Mattila (2015) and Altini\c{s}ik et al. (2016). The sharpness of these lower bounds is assessed numerically and it is conjectured that cn∼5φ−2nc_n\sim 5\varphi^{-2n} as n→∞n\to\infty. In addition, a new closed form expression is derived for the numbers CnC_n, viz. C_n=\frac14 \csc^2\bigg(\frac{\pi}{4n+2}\bigg)=\frac{4n^2}{\pi^2}+\frac{4n}{\pi^2}+\bigg(\frac{1}{12}+\frac{1}{\pi^2}\bigg)+\mathcal{O}\bigg(\frac{1}{n^2}\bigg),\quad n\in\mathbb{Z}_+.
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... Bhatia [9] considered GCD matrices as an example of infinitely divisible matrices. Ilmonen, Haukkanen, and Merikoski [10] instigated the study eigenvalues of meet and join matrices associated with incidence functions. The eigenvalues of meet and join matrices have since been studied thoroughly in recent literature: Mattila and Haukkanen [11] studied positive definiteness and eigenvalues of meet and join matrices, and Mattila, Haukkanen, and Mäntysalo [12] considered the singularity of LCM-type matrices. ...
... To this end, we observe that the matrix E in (1) belongs to the matrix algebra K(n) of all n × n lower triangular 0, 1 matrices such that each main diagonal entry is equal to 1. Clearly every matrix X ∈ K(n) is real and nonsingular and thus XX T is positive definite. We now define the positive constants c n [8] and C n [10] depending only on n such that ...
... However, it has been shown recently that the representative matrices in K(n) corresponding to the constants c n and C n are known a priori. In the paper [10], it has been demonstrated that the symmetric n × n matrix M n satisfying ...
Generalized eigenvalue problems for meet and join matrices on semilattices
Article
  • May 2017
  • LINEAR ALGEBRA APPL
We study generalized eigenvalue problems for meet and join matrices with respect to incidence functions on semilattices. We provide new bounds for generalized eigenvalues of meet matrices with respect to join matrices under very general assumptions. The applied methodology is flexible, and it is shown in the case of GCD and LCM matrices that even sharper bounds can be obtained by applying the known properties of the divisor lattice. These results can also be easily modified for the dual problem of eigenvalues of join matrices with respect to meet matrices, which we briefly consider as well. We investigate the effectiveness of the obtained bounds for select examples involving number-theoretical lattices.
View
... bound for the smallest eigenvalue of certain GCD matrices (see [7,Theorem 4.2]); soon afterwards Ilmonen et al. [11] generalized this result to meet and join matrices. In this article, we show that, under certain circumstances, this method can be extended for the much more general matrix M α,β,γ,δ S,f . ...
... In this article, we show that, under certain circumstances, this method can be extended for the much more general matrix M α,β,γ,δ S,f . The same goes for another method developed by Ilmonen et al.: see [11,Theorem 4.1 and Theorem 6.1]. This is done in Sections 3 and 4. ...
... In Section 5 we turn our attention to the special constants c n originally defined by Hong and Loewy. Currently, no lower bounds are known for this constant for general n, which means that some of the results in [7] and in [11] cannot be applied in practice at all. It turns out that we were able to contribute something to this topic as well, in this article. ...
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... Since their pioneering paper many results on the subject have been published in the literature, see e.g. [1,4,5,10,11,13,15,19,20,21,22]. In that paper Hong and Loewy investigated the asymptotic behavior of the eigenvalues of power GCD matrices by using some tools of number theory. ...
... In 2008, in the light of their MATLAB calculations for n = 2, 3, . . . , 7, Ilmonen, Haukkanen and Merikoski [15] presented an interesting conjecture about the constant c n . Conjecture 1.1. ...
... Conjecture 1.1. [The IHM conjecture, see [15] ...
On a Conjecture of Ilmonen, Haukkanen and Merikoski Concerning the Smallest Eigenvalues of Certain GCD Related Matrices
Article
  • Mar 2016
  • LINEAR ALGEBRA APPL
Let KnK_n be the set of all n×nn\times n lower triangular (0,1)-matrices with each diagonal element equal to 1, Ln={YYT:Y∈Kn}L_n = \{ YY^T: Y\in K_n\} and let cnc_n be the minimum of the smallest eigenvalue of YYTYY^T as Y goes through KnK_n. The Ilmonen-Haukkanen-Merikoski conjecture (the IHM conjecture) states that cnc_n is equal to the smallest eigenvalue of Y0Y0TY_0Y_0^T, where Y0∈KnY_0 \in K_n with (Y0)ij=1−(−1)i+j2(Y_0)_{ij} = \frac{1-(-1)^{i+j}}{2} for i>ji>j. In this paper, we present a proof of this conjecture. In our proof we use an inequality for spectral radii of nonnegative matrices.
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... Korkee [10] defines yet another distinct generalization: a combined meet and join matrix M α,β,γ,δ In Section 5 we turn our attention to the special constants c n originally defined by Hong and Loewy. Currently, no lower bounds are known for this constant for general n, which means that some of the results in [4] and in [7] cannot be applied in practice at all. It turns out that we were able to contribute something to this topic as well, in this article. ...
... for all x, y ∈ P . We also adopt one constant c n from Hong and Loewy [4] and another C n from Ilmonen et al. [7]. Let K(n) denote the set of all n × n lower triangular 0, 1 matrices with each main diagonal element equal to 1. Now for every positive integer n we define c n = min{λ X ∈ K(n) and λ is the smallest eigenvalue of XX T } and C n = max{λ X ∈ K(n) and λ is the largest eigenvalue of XX T }. ...
... The proof is similar to the proof of Theorem 3.1. [7] follows directly from Theorem 3.2. In this case α = 0, and therefore f does not need to be semimultiplicative, nor does (P, ≺) need to be a meet semilattice with0 as the smallest element. ...
On the eigenvalues of combined meet and join matrices
Article
  • Mar 2014
  • LINEAR ALGEBRA APPL
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 cnc_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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... 3 [10] [12] = 2 30 3 60 , while [2 5 3 10 , 2 6 ] e does not exist. Note that (m, n) e exists if and only if [m, n] e exists. ...
... f ) ij = f (x i ∨ x j ). Rajarama Bhat [15] and Haukkanen [6] introduced meet matrices and Korkee and Haukkanen [12] introduced join matrices. Explicit formulae for the determinant and the inverse of meet and join matrices are presented in [6] [11] [12] [15] (see also [2] [8] [17]). ...
... Rajarama Bhat [15] and Haukkanen [6] introduced meet matrices and Korkee and Haukkanen [12] introduced join matrices. Explicit formulae for the determinant and the inverse of meet and join matrices are presented in [6] [11] [12] [15] (see also [2] [8] [17]). Most of these formulae are presented on meet-closed sets S (i.e., x i , x j ∈ S ⇒ x i ∧ x j ∈ S) and joinclosed sets S (i.e., x i , x j ∈ S ⇒ x i ∨ x j ∈ S). ...
Meet and join matrices in the poset of exponential divisors
Article
Full-text available
  • Jun 2009
It is well-known that (ℤ+, |) = (ℤ+, GCD, LCM) is a lattice, where | is the usual divisibility relation and GCD and LCM stand for the greatest common divisor and the least common multiple of positive integers. The number d=∏k=1rpkd(k) d = \prod\nolimits_{k = 1}^r {p_k^{d^{(k)} } } d = \prod\nolimits_{k = 1}^r {p_k^{d^{(k)} } } is said to be an exponential divisor or an e-divisor of n=∏k=1rpkn(k) n = \prod\nolimits_{k = 1}^r {p_k^{n^{(k)} } } n = \prod\nolimits_{k = 1}^r {p_k^{n^{(k)} } } (n > 1), written as d | e n, if d (k) for all prime divisors p k of n. It is easy to see that (ℤ+\{1}, | e is a poset under the exponential divisibility relation but not a lattice, since the greatest common exponential divisor (GCED) and the least common exponential multiple (LCEM) do not always exist. In this paper we embed this poset in a lattice. As an application we study the GCED and LCEM matrices, analogues of GCD and LCM matrices, which are both special cases of meet and join matrices on lattices.
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... Then, in the light of their MATLAB calculations for n = , , . . . , , Ilmonen, Haukkanen and Merikoski [8] ...
... In this paper, introducing a new constant cn(a), we have expanded the results on the nding all minimizing matrices of the constant cn and its asymptotic behaviour to a larger class of matrices. We do not reckon that our constant cn(a) could be used in eigenvalue estimation of GCD and related matrices as cn was used in the literature, see [3,6,8,9,11,12]. However, it seems possible that the techniques of this paper could be applied to a larger class of matrices than those considered in this present paper. ...
On the smallest singular value in the class of unit lower triangular matrices with entries in [−a, a]
Article
Full-text available
  • Jun 2021
Given a real number a ≥ 1, let Kn(a) be the set of all n × n unit lower triangular matrices with each element in the interval [−a, a]. Denoting by λn(·) the smallest eigenvalue of a given matrix, let cn(a) = min {λ n(YYT) : Y ∈ Kn(a)}. Then cn(a) is the smallest singular value in Kn(a). We find all minimizing matrices. Moreover, we study the asymptotic behavior of cn(a) as n → ∞. Finally, replacing [−a, a] with [a, b], a ≤ 0 < b, we present an open question: Can our results be generalized in this extension?
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... The study of GCD matrices began already in 1876, when Smith [1] studied the determinant of the n×n matrix having the greatest common divisor of i and j as its ij element. The properties of meet and join matrices on semilattices have been studied in recent years by several authors: Ilmonen, Haukkanen, and Merikoski [2] studied the eigenvalues of meet and join matrices associated with incidence functions. Mattila and Haukkanen [3] studied positive definiteness and eigenvalues of meet and join matrices, and Mattila, Haukkanen, and Mäntysalo [4] considered the singularity of LCM-type matrices. ...
... For d even, this submatrix exists trivially. For d odd, one uses the fact that S ∨⌊d/2⌋ ⊂ S ∨⌈d/2⌉ since any element x i 1 ∨ · · · ∨ x i d ∈ S ∨⌊d/2⌋ is also an element x i 1 ∨ · · · ∨ x i ⌊d/2⌋ ∨ x i k ∈ S ∨⌈d/2⌉ for any 1 ≤ k ≤ ⌊d/2⌋.2 Sylvester's rank inequality: rank(AB) ≥ rank A + rank B − n for any k × n matrix A and n × m matrix B. ...
On the structure of join tensors with applications to tensor eigenvalue problems
Article
  • May 2017
We investigate the structure of join tensors, which may be regarded as the multivariable extension of lattice-theoretic join matrices. Explicit formulae for a polyadic decomposition (i.e., a linear combination of rank-1 tensors) and a tensor-train decomposition of join tensors are derived on general join semilattices. We discuss conditions under which the obtained decompositions are optimal in rank, and examine numerically the storage complexity of the obtained decompositions for a class of LCM tensors as a special case of join tensors. In addition, we investigate numerically the sharpness of a theoretical upper bound on the tensor eigenvalues of LCM tensors.
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... Since the paper of Hong and Loewy many results on the asymptotic behavior of the eigenvalues of the GCD and related matrices have been published in the literature, see [3,4,11,12,14]. In 2008 Ilmonen, Haukkanen and Merikoski [17] examined the eigenvalues of certain abstract generalizations of the GCD matrix and the LCM matrix on posets. Then Mattila and Haukkanen [25] and Mattila [22] gave new bounds for the eigenvalues of such abstract generalizations. ...
... In the study of eigenvalues of GCD, LCM and related matrices, some authors [3,11,12,13,14] investigated asymptotic behavior of the eigenvalues of such matrices defined on S = {x 1 , x 2 ,... ,x n } and they gave lower bounds for the smallest eigenvalues and upper bounds for the largest eigenvalues of such matrices. In addition, authors of [17,22,23,24,25] obtained such bounds on some certain constants and particular arithmetical functions by using matrix theoretic techniques. In this paper, we have obtained not only a lower (an upper) bound but also an upper (a lower) bound for the smallest (largest) eigenvalue of the GCD matrix as well as the LCM matrix defined on S n by using a different technique from above papers. ...
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... More recently, Hong and Enoch Lee [18] investigated the asymptotic behavior of the eigenvalues of the reciprocal LCM matrix ( 1 [x i ,x j ] r ) for a positive real number r. Also, Ilmonen, Haukkanen and Merikoski [21] examined the eigenvalues of certain abstract generalizations of (S f ) and [S f ] on posets. Now let C = {1, 2, . . . ...
... Also, for ε > 1, ζ(ε) 2 ζ(2ε) is a positive real number. Thus, the proof of (i) follows from (21). Now let us consider the matrix (C Jε ) for ε > 2. By Theorem 11(ii) and (10) 1 ...
A proof of a conjecture on monotonic behavior of the smallest and the largest eigenvalues of a number theoretic matrix
Article
  • Apr 2015
  • LINEAR ALGEBRA APPL
In this study we investigate the monotonic behavior of the smallest eigenvalue and the largest eigenvalue of the matrix , where the ij-entry of is 1 if and 0 otherwise. We present a proof of the Mattila-Haukkanen conjecture which states that for every , and . Also, we prove that and and we give a lower bound for .
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... is positive definite and lim n→∞ λ (q) n ≥ 0 for a given arbitrary integer q ≥ 1. On a different point of view, in 2008, the authors [14] examined the eigenvalues of certain abstract generalizations of the GCD matrix and the LCM matrix on posets. Recently, Mattila and Haukkanen [21] and Mattila [18] gave new bounds for the eigenvalues of such abstract generalizations. ...
... In the study of eigenvalues of GCD, LCM and related matrices, some authors [2,9,10,11,12] investigated asymptotic behavior of the eigenvalues of such matrices and also they gave lower bounds for the smallest eigenvalue and upper bounds for the largest eigenvalue of these matrices. On the other hand, some other authors [14,18,19,20,21] obtained such lower and upper bounds for the eigenvalues of these matrices by using matrix theoretic techniques. In these papers, the authors obtained lower bounds for the smallest eigenvalue of GCD-related matrices depending on some certain constants on the eigenvalues of particular matrices and some values of particular arithmetical functions. ...
On Bounds for the Smallest and the Largest Eigenvalues of GCD and LCM Matrices
Article
Full-text available
  • Jan 2016
  • MATH INEQUAL APPL
In this paper, improving a famous result of Wolkowicz and Styan for the GCD matrix (Sn)(S_n) and the LCM matrix [Sn][S_n] defined on Sn={1,2,…,n}S_n=\{1,2,\ldots,n\}, we present new upper and lower bounds for the smallest and the largest eigenvalues of (Sn)(S_n) and [Sn][S_n] in terms of particular arithmetical functions.
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... The LCM reciprocal GCD matrix on S with respect to f is defined analogously. Some results on eigenvalues, norms, determinants and inverses of certain special cases of GCD reciprocal LCM matrices have been published in the literature (see12131419,24,28,3031323341,43]) and some results on determinants and inverses of certain special cases of unitary analogs of GCD reciprocal LCM matrices also have been published in the literature (see [28,29,30,40]). Wintner [43] and Lindqvist and Seip [24] apply deep analytic tools to obtain results on eigenvalues of the GCD reciprocal LCM matrix [(i, j) /[i, j] ]. ...
... Wintner [43] and Lindqvist and Seip [24] apply deep analytic tools to obtain results on eigenvalues of the GCD reciprocal LCM matrix [(i, j) /[i, j] ]. Ilmonen et al. [19] note that latticetheoretic methods reveal some properties of eigenvalues of certain general meet and join matrices, including meet reciprocal join matrices. Korkee [20] considers determinants and inverses of meet reciprocal join matrices in a lattice-theoretic level but he does not consider the unitary case. ...
On unitary analogs of GCD reciprocal LCM matrices
Article
Full-text available
  • Jul 2010
A divisor d + of n + is said to be a unitary divisor of n if (d, n/d) = 1. In this article we examine the greatest common unitary divisor (GCUD) reciprocal least common unitary multiple (LCUM) matrices. At first we concentrate on the difficulty of the non-existence of the LCUM and we present three different ways to overcome this difficulty. After that we calculate the determinant of the three GCUD reciprocal LCUM matrices with respect to certain types of functions arising from the LCUM problematics. We also analyse these classes of functions, which may be referred to as unitary analogs of the class of semimultiplicative functions, and find their connections to rational arithmetical functions. Our study shows that it does make a difference how to extend the concept of LCUM.
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... Since that, lots of results concerning the determinants and related topics of GCD matrices, least common multiple (LCM) matrices, meet matrices and join matrices have been published in the literature (see, e.g. [2,3,5,9,10,13,15,16,18,27]). ...
... Since then, many results concerning determinants and other characteristics of GCD and related matrices have been published in the literature (see, e.g. [2,3,5,9,10,13,15,16,18,27]). We next review some generalizations and analogues of GCD matrices. ...
Smith meets Smith: Smith normal form of Smith matrix
Article
Full-text available
  • May 2011
In 1861, Henry John Stephen Smith [H.J.S. Smith, On systems of linear indeterminate equations and congruences, Philos. Trans. Royal Soc. London. 151 (1861), pp. 293–326] published famous results concerning solving systems of linear equations. The research on Smith normal form and its applications started and continues. In 1876, Smith [H.J.S. Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7 (1875/76), pp. 208–212] calculated the determinant of the n × n matrix ((i, j)), having the greatest common divisor (GCD) of i and j as its ij entry. Since that, many results concerning the determinants and related topics of GCD matrices, LCM matrices, meet matrices and join matrices have been published in the literature. In this article these two important research branches developed by Smith, in 1861 and in 1876, meet for the first time. The main purpose of this article is to determine the Smith normal form of the Smith matrix ((i, j)). We do this: we determine the Smith normal form of GCD matrices defined on factor closed sets.
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... More recently, Hong and Enoch Lee [18] investigated the asymptotic behavior of the eigenvalues of the reciprocal LCM matrix 1 [x i ,x j ] r for a positive real number r. Also, Ilmonen et al. [21] examined the eigenvalues of certain abstract generalizations of (S f ) and [S f ] on posets. Now let C = {1, 2, . . . ...
... Also, for ε > 1, ζ(ε) 2 ζ(2ε) is a positive real number. Thus, the proof of (i) follows from (21). Now let us consider the matrix (C Jε ) for ε > 2. By Theorem 11(ii) and (10) 1 λ (1) m (C Jε ) ...
On inverses of GCD matrices associated with multiplicative functions and a proof of the Hong–Loewy conjecture
Article
  • Feb 2009
  • LINEAR ALGEBRA APPL
Let C={1,2,…,m} and f be a multiplicative function such that (f∗μ)(k)>0 for every positive integer k and the Euler product converges. Let (Cf)=(f(i,j)) be the m×m matrix defined on the set C having f evaluated at the greatest common divisor (i,j) of i and j as its ij-entry. In the present paper, we first obtain the least upper bounds for the ij-entry and the absolute row sum of any row of (Cf)-1, the inverse of (Cf), in terms of ζf. Specializing these bounds for the arithmetical functions f=Nε,Jε and σε we examine the asymptotic behavior the smallest eigenvalue of each of matrices (CNε),(CJε) and (Cσε) depending on ε when m tends to infinity. We conclude our paper with a proof of a conjecture posed by Hong and Loewy [S. Hong, R. Loewy, Asymptotic behavior of eigenvalues of greatest common divisor matrices, Glasg. Math. J. 46 (2004) 551–569].
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... The properties of meet and join matrices have been studied by many authors (see, e.g., [3,8,10,15,16,18,20,21,23,25,27,28]). Haukkanen [8] calculated the determinant of (S) f on an arbitrary set S and obtained the inverse of (S) f on a lower-closed set S and Korkee and Haukkanen [17] obtained the inverse of (S) f on a meet-closed set S. Korkee and Haukkanen [18] present, among others, formulas for the determinant and inverse of [S] f on meet-closed, join-closed, lower-closed and upper-closed sets S. ...
... In this section we give a factorization of the matrix [X, Y ] f = (f (x i ∨ y j )). A large number of similar factorizations is presented in the literature, for example in [16] the matrix [S] f is factorized in case when S is join-closed. The idea of this kind of factorization may be considered to originate from Pólya and Szegö [26]. ...
Determinant and inverse of join matrices on two sets
Article
Full-text available
  • Oct 2011
  • LINEAR ALGEBRA APPL
Let (P,⪯)(P,\preceq) 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 P by (X,Y)f=(f(xi∧yj))(X,Y)_f=(f(x_i\wedge y_j)) and [X,Y]f=(f(xi∨xj))[X,Y]_f=(f(x_i\vee x_j)) respectively. Here we present expressions for the determinant and the inverse of [X,Y]f[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[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(X,Y)_f and the usual meet and join matrices (S)f(S)_f and [S]f[S]_f. We also apply these formulas to LCM, MAX, GCD and MIN matrices, which are special cases of join and meet matrices.
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... Hong and Loewy [20] examined the asymptotic behavior of eigenvalues of power GCD matrices. Ilmonen, Haukkanen, and Merikoski [22] examined the eigenvalues of meet and join matrices with respect to f. Recently Mattila and Haukkanen [32] studied positive definiteness and eigenvalues of meet and join matrices, and Mattila, Haukkanen, and Mäntysalo [33] considered singularity of LCM-type matrices. ...
Computation of extremal eigenvalues of high-dimensional lattice-theoretic tensors via tensor-train decompositions
Article
Full-text available
  • May 2017
This paper lies in the intersection of several fields: number theory, lattice theory, multilinear algebra, and scientific computing. We adapt existing solution algorithms for tensor eigenvalue problems to the tensor-train framework. As an application, we consider eigenvalue problems associated with a class of lattice-theoretic meet and join tensors, which may be regarded as multidimensional extensions of the classically studied meet and join matrices such as GCD and LCM matrices, respectively. In order to effectively apply the solution algorithms, we show that meet tensors have an explicit low-rank tensor-train decomposition with sparse tensor-train cores with respect to the dimension. Moreover, this representation is independent of tensor order, which eliminates the so-called curse of dimensionality from the numerical analysis of these objects and makes the solution of tensor eigenvalue problems tractable with increasing dimensionality and order. For LCM tensors it is shown that a tensor-train decomposition with an a priori known TT-rank exists under certain assumptions. We present a series of easily reproducible numerical examples covering tensor eigenvalue and generalized eigenvalue problems that serve as future benchmarks. The numerical results are used to assess the sharpness of existing theoretical estimates.
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... Bhatia [12] investigated the general infinitely divisible matrices. Ilmonen et al. [13] gave lower bounds for the smallest eigenvalues of certain positive definite meet and join matrices with respect f , and estimated the eigenvalues of complex valued meet and join matrices. ...
On meet hypermatrices and their eigenvalues
Article
  • Jun 2015
Let be a locally finite meet semilattice. Letbe a finite subset of , and let be a complex-valued function on . The -dimensional hypermatrix of order , , given byis called the order meet hypermatrix on with respect to . We consider -dimensional meet hypermatrices of order . As an example, we consider GCD hypermatrices. We examine the structure of order meet hypermatrices with respect to , and provide a structure theoretical result that is a generalization of a known result for meet matrices. We also give a region in which all the eigenvalues of an -dimensional order meet hypermatrix with respect to a real-valued lie, and using that we obtain results concerning positive definiteness and E-eigenvalues of meet hypermatrices. Characteristics of meet matrices and the eigenvalues of supersymmetric hypermatrices are under active research, but the eigenvalues of GCD and related hypermatrices have not hitherto been considered in the literature.
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... Let α, β ∈ R. Our goal here is to find bounds for the eigenvalues of the n × n matrix having (i, j) α [i, j] β as its ij entry. In order to do this we use similar techniques as Ilmonen et al [15] and Hong and Loewy [13]. One of the methods may be considered to originate from Hong and Loewy [12]. ...
On the eigenvalues of certain number-theoretic matrix
Article
Full-text available
  • Sep 2013
In this paper we study the structure and give bounds for the eigenvalues of the n×nn\times n matrix, which ij entry is (i,j)α[i,j]β(i,j)^\alpha[i,j]^\beta, where \alpha,\beta\in\Rset, (i,j) is the greatest common divisor of i and j and [i,j] is the least common multiple of i and j. Currently only O-estimates for the greatest eigenvalue of this matrix can be found in the literature, and the asymptotic behaviour of the greatest and smallest eigenvalue is known in case when α=β\alpha=\beta.
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... Hong and Loewy [26,27] studied the asymptotic behaviour of the eigenvalues of the matrix (f (S n )) and Hong [24] gives lower bound for its eigenvalues in a case when d | x i for some x i ∈ S ⇒ (f * µ)(d) > 0 as well as continues the research on the asymptotic behaviour of the eigenvalues. Altinisik [3] provides information about the eigenvalues of GCD matrices, a paper by Hong and Lee [25] addresses the eigenvalues of reciprocal power LCM matrices and there is also one paper about the eigenvalues of meet and join matrices by Ilmonen et al. [30]. ...
On the positive definiteness and eigenvalues of meet and join matrices
Article
  • Sep 2012
  • DISCRETE MATH
In this paper we study the positive definiteness of meet and join matrices using a novel approach. When the set SnS_n is meet closed, we give a sufficient and necessary condition for the positive definiteness of the matrix (Sn)f(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 SnS_n we are able to reduce the assumptions on the function f while still preserving the positive definiteness of the matrix (Sn)f(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.
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... The structure of an LCM matrix [i, j] n×n is the following (I. Korkee, P. Haukkanen [10]) [i, j] n×n ...
Generalized LCM matrices
Article
  • Aug 2011
  • PUBL MATH-DEBRECEN
Let f be an arithmetical function. The matrix [f[i,j]]n×n[f[i,j]]_{n\times n} given by the value of f in least common multiple of [i,j], f([i,j])f\big([i,j]\big) as its i,  ji,\; j entry is called the least common multiple (LCM) matrix. We consider the generalization of this matrix where the elements are in the form f(n,[i,j])f\big(n,[i,j]\big) and f(n,i,j,[i,j])f\big(n,i,j,[i,j]\big).
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Bounds on the spectrum of nonsingular triangular (0,1)-matrices
Article
  • Feb 2021
Let Kn be the set of all nonsingular n×n lower triangular (0,1)-matrices. Hong and Loewy (2004) introduced the numberscn=min{λ|λis an eigenvalue ofXXT,X∈Kn},n∈Z+. A related family of numbers was considered by Ilmonen, Haukkanen, and Merikoski (2008):Cn=max{λ|λis an eigenvalue ofXXT,X∈Kn},n∈Z+. These numbers can be used to bound the singular values of matrices belonging to Kn and they appear, e.g., in eigenvalue bounds for power GCD matrices, lattice-theoretic meet and join matrices, and related number-theoretic matrices. In this paper, it is shown that for n odd, one has the lower boundcn≥1125φ−4n+225φ−2n−255nφ−2n−2325+n+225φ2n+255nφ2n+125φ4n, and for n even, one hascn≥1125φ−4n+425φ−2n−255nφ−2n−25+n+425φ2n+255nφ2n+125φ4n, where φ denotes the golden ratio. These lower bounds improve the estimates derived previously by Mattila (2015) and Altınışık et al. (2016). The sharpness of these lower bounds is assessed numerically and it is conjectured that cn∼5φ−2n as n→∞. In addition, a new closed form expression is derived for the numbers Cn, viz.Cn=14csc2⁡(π4n+2)=4n2π2+4nπ2+(112+1π2)+O(1n2),n∈Z+.
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A proof of a conjecture on monotonic behavior of the largest eigenvalue of a number-theoretic matrix
Article
  • Mar 2015
In this study we investigate the monotonic behavior of the largest eigenvalue of the n×n matrix EnTEn, where the i j- entry of En is 1 if j|i and 0 otherwise and hence we present a proof of a part of the Mattila-Haukkanen conjecture [16]. MSC2010. 15A18, 15A42, 11A25
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The Principle and Algorithm for Generating Incidence Matrix for Any Arbitrary Network
Chapter
  • May 2013
In this paper, a new principle and algorithm for obtaining the incidence matrix for any arbitrary network which were represented by nodes and segments while we have already known the endpoints of each line segments in 2D space were introduced. In addition, a calculated procedure was compiled by C++ language and two extra examples were calculated. The results shown that the principal and algorithm we stated were right for auto-generating of the incidence matrix for any arbitrary network.
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Determinant and Inverse of Meet and Join Matrices
Article
Full-text available
  • May 2007
We define meet and join matrices on two subsets X and Y of a lattice (P,≼) with respect to a complex-valued function f on P by (X,Y)f=(f(xi∧yi)) and [X,Y]f=(f(xi∨yi)), respectively. We present expressions for the determinant and the inverse of (X,Y)f and [X,Y]f, and as special cases we obtain several new and known formulas for the determinant and the inverse of the usual meet and join matrices (S)f and [S]f.
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An upper bound for the ℒ P norm of a GCD-related matrix
Article
Full-text available
  • Jan 2006
  • J INEQUAL APPL
We find an upper bound for the ℓ p norm of the n × n matrix whose ij entry is ( i , j ) s / [ i , j ] r , where ( i , j ) and [ i , j ] are the greatest common divisor and the least common multiple of i and j and where r and s are real numbers. In fact, we show that if r > 1 / p and s < r − 1 / p , then ‖ ( ( i , j ) s / [ i , j ] r ) n × n ‖ p < ζ ( r p ) 2 / p ζ ( r p − s p ) 1 / p / ζ ( 2 r p ) 1 / p for all positive integers n , where ζ is the Riemann zeta function.
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GCD matrices, posets, and nonintersecting paths
Article
Full-text available
  • Jul 2004
We show that with any finite partially ordered set P (which need not be a lattice) one can associate a matrix whose determinant factors nicely. This was also noted by D. A. Smith [J. Reine Angew. Math. 251, 100–109 (1971; Zbl 0224.06002)], although his proof uses manipulations in the incidence algebra of P while ours is combinatorial, using nonintersecting paths in a digraph. As corollaries, we obtain new proofs for and generalizations of a number of results in the literature about GCD matrices and their relatives.
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Some analogues of Smith's determinant
Article
Full-text available
  • Nov 1996
We calculate the determinants of the greatest common divisor (GCD) and the least common multiple (LCM) matrices associated with an arithmetical function on gcd-closed and lcm-closed sets. We also consider some unitary analogues of these determinants.
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Note on some greatest common divisor matrices
Article
Full-text available
  • Jan 1998
Some quadratic forms related to "greatest common divisor matrices" are represented in terms of L 2 -norms of rather simple functions. Our formula is especially useful when the size of the matrix grows, and we will study the asymptotic behaviour of the smallest and largest eigenvalues. Indeed, a sharp bound in terms of the zeta function is obtained. Our leading example is a hybrid between Hilbert's matrix and Smith's matrix.
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Some Characterizations of Totients
Article
Full-text available
  • Jan 1996
An arithmetical function is said to be a totient if it is the Dirichlet convolution between a completely multiplicative function and the inverse of a completely multiplicative function. Euler's phi-function is a famous example of a totient. All completely multiplicative functions are also totients. There is a large number of characterizations of completely multiplicative functions in the literature, while characterizations of totients have not been widely studied in the literature. In this paper we present several arithmetical identities serving as characterizations of totients. We also introduce a new concrete example of a totient.
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Notes on the divisibility of GCD and LCM matrices
Article
Full-text available
  • May 2005
Let S={x1,x2,…,xn} be a set of positive integers, and let f be an arithmetical function. The matrices (S)f=[f(gcd(xi,xj))] and [S]f=[f(lcm [xi,xj])] are referred to as the greatest common divisor (GCD) and the least common multiple (LCM) matrices on S with respect to f, respectively. In this paper, we assume that the elements of the matrices (S)f and [S]f are integers and study the divisibility of GCD and LCM matrices and their unitary analogues in the ring Mn(ℤ) of the n×n matrices over the integers.
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Determinant and Inverse of Meet and Join Matrices
Article
Full-text available
  • May 2007
We define meet and join matrices on two subsets X and Y of a lattice (P,≼) with respect to a complex-valued function f on P by (X,Y)f=(f(xi∧yi)) and [X,Y]f=(f(xi∨yi)), respectively. We present expressions for the determinant and the inverse of (X,Y)f and [X,Y]f, and as special cases we obtain several new and known formulas for the determinant and the inverse of the usual meet and join matrices (S)f and [S]f.
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GCD matrices, posets, and nonintersecting paths
Article
Full-text available
  • Jul 2004
We show that with any finite partially ordered set P (which need not be a lattice) one can associate a matrix whose determinant factors nicely. This was also noted by D. A. Smith [J. Reine Angew. Math. 251, 100–109 (1971; Zbl 0224.06002)], although his proof uses manipulations in the incidence algebra of P while ours is combinatorial, using nonintersecting paths in a digraph. As corollaries, we obtain new proofs for and generalizations of a number of results in the literature about GCD matrices and their relatives.
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On Smith's Determinant
Article
  • Jun 1997
  • LINEAR ALGEBRA APPL
We give a brief review of papers relating to Smith's determinant and point out a common structure that can be found in many extensions and analogues of Smith's determinant. We present the common structure in the language of posets. We also make an investigation on a conjecture of Beslin and Ligh on greatest common divisor (GCD) matrices in the sense of meet matrices and give characterizations of the posets satisfying the conjecture. Further, we give a counterexample for the conjecture of Bourque and Ligh that the least common multiple matrix on any GCD-closed set is invertible.
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Greatest common divisor matrices
Article
  • Jun 1989
  • LINEAR ALGEBRA APPL
Let S={x1, x2,…, xn} be a set of distinct positive integers. Then n × n matrix [S]=(Sij), where Sij=(xi, xj), the greatest common divisor of xi and xj, is call the greatest common divisor (GCD) matrix on S. We initiate the study of GCD matrices in the direction of their structure, determinant, and arithmetic in Zn. Several open problems are posed.
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On GCD and LCM matrices
Article
  • Sep 1992
  • LINEAR ALGEBRA APPL
Let S = {x1, x2,…, xn} be a set of distinct positive integers. The matrix (S) having the greatest common divisor (xi, xj) of xi and xj as its i, j entry is called the greatest common divisor (GCD) matrix on S. The matrix [S] having the least common multiple of xi and xj as its i, j entry is called the least common multiple (LCM) matrix on S. The set S is factor-closed if it contains every divisor of each of its elements. If S is factor-closed, we calculate the inverses of the GCD and LCM matrices on S and show that [S](S)−1 is an integral matrix. We also extend a result of H. J. S. Smith by calculating the determinant of [S] when (xi, xj)∈ S for 1 ⩽ i, j ⩽ n.
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Factorization of matrices associated with classes of arithmetical functions
Article
  • Jan 2003
  • Colloq Math
Let f be an arithmetical function. A set S = {x1,…,xn} of n distinct positive integers is called multiple closed if y ∈ S whenever x |y| lcm(S) for any x ∈ S, where lcm(S) is the least common multiple of all elements in S. We show that for any multiple closed set S and for any divisor chain S (i.e. x1| … |xn), if f is a completely multiplicative function such that (f * μ)(d) is a nonzero integer whenever d | lcm(S), then the matrix (f(xi, xj)) having f evaluated at the greatest common divisor (xi, xj) of xi and xj as its i, j-entry divides the matrix (f [xi, xj]) having f evaluated at the least common multiple [xi, xj ] of xi and xj as its i, j-entry in the ring Mn(ℤ) of n × n matrices over the integers. But such a factorization is no longer true if f is multiplicative.
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Introduction to Arithmetical Functions
Article
  • Jan 1986
The theory of arithmetical functions has always been one of the more active parts of the theory of numbers. The large number of papers in the bibliography, most of which were written in the last forty years, attests to its popularity. Most textbooks on the theory of numbers contain some information on arithmetical functions, usually results which are classical. My purpose is to carry the reader beyond the point at which the textbooks abandon the subject. In each chapter there are some results which can be described as contemporary, and in some chapters this is true of almost all the material. This is an introduction to the subject, not a treatise. It should not be expected that it covers every topic in the theory of arithmetical functions. The bibliography is a list of papers related to the topics that are covered, and it is at least a good approximation to a complete list within the limits I have set for myself. In the case of some of the topics omitted from or slighted in the book, I cite expository papers on those topics.
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Matrices associated with multiplicative functions
Article
  • Feb 1995
  • LINEAR ALGEBRA APPL
Let f be a multiplicative function and S = {x1, x2, …, xn} a set of distinct positive integers. Denote by (f[xi, xj]) the n × n matrix having f evaluated at the least common multiple [xi, xj] of xi and xj as its i, j entry. If S is factor-closed, we calculate the determinant of this matrix and (if it is invertible) its inverse, and show that for a certain class of functions the n × n matrix (f(xi, xj)) having f evaluated at the greatest common divisor of xi and xj as its, i, j entry is a factor of the matrix (f[xi, xj]) in the ring of n × n matrices over the integers. We also determine conditions on f that will guarantee the matrix (f[xi, xj]) is positive definite.
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On greatest common divisor matrices and their applications
Article
  • Nov 1991
  • LINEAR ALGEBRA APPL
Let S = {x1, x2,… xn} be a set of distinct positive integers. The n × n matrix [S] = ((sij)), where sij = (xi, xj), the greatest common divisor of xi and xj, is called the greatest common divisor (GCD) matrix on S. We study the structure of a GCD matrix and obtain interesting relations between its determinant. Euler's totient function, and Moebius function. We also determine some arithmetic progressions related to GCD matrices. Then we generalize the results to general partially ordered sets and show a variety of applications.
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Matrices associated with arithmetical functions
Article
  • May 1993
Let f be an arithmetical function and S={x1,x2,…,xn} a set of distinct positive integers. Denote by [f(xi,xj}] the n×n matrix having f evaluated at the greatest common divisor (xi,xj) of xi, and xj as its ij-entry. We will determine conditions on f that will guarantee the matrix [f(xi,xj)] is positive definite and, in fact, has properties similar to the greatest common divisor (GCD) matrix [(xi,xj)] where f is the identity function. The set S is gcd-closed if (xi,xj)∈S for 1≤ ij ≤ n. If S is gcd-closed, we calculate the determinant and (if it is invertible) the inverse of the matrix [f(xi,xj)]. Among the examples of determinants of this kind are H. J. S. Smith's determinant det[(i,j)].
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Bounds for determinants of meet matrices associated with incidence functions
Article
  • May 2001
  • LINEAR ALGEBRA APPL
We consider meet matrices on meet-semilattices as an abstract generalization of greatest common divisor (gcd) matrices. Some new bounds for the determinant of meet matrices and a formula for the inverse of meet matrices are given.
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On the factorization of LCM matrices on gcd-closed sets
Article
  • Apr 2002
  • LINEAR ALGEBRA APPL
Let S={x1,…,xn} be a set of n distinct positive integers. The matrix having the greatest common divisor (GCD) (xi,xj) of xi and xj as its i,j-entry is called the greatest common divisor matrix, denoted by (S)n. The matrix having the least common multiple (LCM) [xi,xj] of xi and xj as its i,j-entry is called the least common multiple matrix, denoted by [S]n. The set is said to be gcd-closed if (xi,xj)∈S for all 1⩽i,j⩽n. In this paper we show that if n⩽3, then for any gcd-closed set S={x1,…,xn}, the GCD matrix on S divides the LCM matrix on S in the ring Mn(Z) of n×n matrices over the integers. For n⩾4, there exists a gcd-closed set S={x1,…,xn} such that the GCD matrix on S does not divide the LCM matrix on S in the ring Mn(Z). This solves a conjecture raised by the author in 1998.
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An analysis of GCD and LCM matrices via the LDL^T-factorization
Article
  • Jan 2004
  • ELECTRON J LINEAR AL
Let S = {x1 ,x 2 ,...,x n} be a set of distinct positive integers such that gcd(xi ,x j ) ∈ S for 1 ≤ i, j ≤ n. Such a set is called GCD-closed. In 1875/1876, H.J.S. Smith showed that, if the set S is "factor-closed", then the determinant of the matrix eij =g cd(xi ,x j )i s det(E )= n m=1 φ(xm), where φ denotes Euler's Phi-function. Since the early 1990's there has been a rebirth of interest in matrices defined in terms of arithmetic functions defined on S. In 1992, Bourque and Ligh conjectured that the matrix fij =l cm(xi ,x j ) is nonsingular. Several authors have shown that, although the conjecture holds for n ≤ 7, it need not hold in general. At present there are no known necessary conditions for F to be nonsingular, but many have offered sufficient conditions. In this note, a simple algorithm is offered for computing the LDLT -Factorization of any matrix bij = f (gcd(xi ,x j)), where f : S → C. This factorization gives us an easy way of answering the question of singularity, computing its determinant, and determining its inertia (the number of positive negative and zero eigenvalues). Using this factorization, it is argued that E is positive definite regardless of whether or not S is GCD-closed (a known result), and that F is indefinite for n ≥ 2. Also revisited are some of the known sufficient conditions for the invertibility of F , which are justified in the present framework, and then a few new sufficient conditions are offered. Similar statements are made for the reciprocal matrices gij =g cd(xi ,x j )/lcm(xi ,x j )a ndhij =l cm(xi ,x j)/ gcd(xi ,x j ).
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Asymptotic behavior of eigenvalues of greatest common divisor matrices
Article
  • Sep 2004
  • GLASGOW MATH J
Let {xi}i=1∞\{x_i\}_{i=1}^{\infty} be an arbitrary strictly increasing infinite sequence of positive integers. For an integer n≥1n\ge 1, let Sn={x1,…,xn}S_n=\{x_1,\ldots,x_n\}. Let ε\varepsilon be a real number and q≥1q\ge 1 a given integer. Let \smash{λn(1)≤⋯≤λn(n)\lambda _n^{(1)}\le \cdots\le \lambda _n^{(n)}} be the eigenvalues of the power GCD matrix ((xi,xj)ε)((x_i, x_j)^{\varepsilon}) having the power (xi,xj)ε(x_i,x_j)^{\varepsilon} of the greatest common divisor of xix_i and xjx_j as its i,j-entry. We give a nontrivial lower bound depending on x1x_1 and n for \smash{λn(1)\lambda _n^{(1)}} if ε>0\varepsilon>0. Especially for ε>1\varepsilon>1, this lower bound is given by using the Riemann zeta function. Let x≥1x\ge 1 be an integer. For a sequence \smash{{xi}i=1∞\{x_i\}_{i=1}^{\infty }} satisfying that (xi,xj)=x(x_i, x_j)=x for any i≠ji\ne j and \smash{∑i=1∞1xi=∞\sum_{i=1}^{\infty }{1\over {x_i}}=\infty}, we show that if 0<ε≤10<\varepsilon\le 1, then \smash{limn→∞λn(1)=x1ε−xε{\rm lim}_{n\rightarrow \infty }\lambda _n^{(1)}=x_1^{\varepsilon}-x^{\varepsilon }}. Let a≥0,b≥1a\ge 0, b\ge 1 and e≥0e\ge 0 be any given integers. For the arithmetic progression \smash{{xi−e+1=a+bi}i=e∞\{x_{i-e+1}=a+bi\}_{i=e}^{\infty}}, we show that if 0<ε≤10<\varepsilon\le 1, then \smash{limn→∞λn(q)=0{\rm lim}_{n\rightarrow \infty }\lambda _n^{(q)}=0}. Finally, we show that for any sequence \smash{{xi}i=1∞\{x_i\}_{i=1}^{\infty}} and any \smash{ε>0\varepsilon>0, λn(n−q+1)\lambda_n^{(n-q+1)}} approaches infinity when n goes to infinity.
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Lower bounds for determinants of matrices associated with classes of arithmetical Functions
Article
  • Apr 1999
  • LINEAR ALGEBRA APPL
Let be an arithmetical function and S = x1, xn a set of distinct positive integers. Let ((xi,xj)) denote the n × n matrix having evaluated at the greatest common divisor of and as its entry and denote the matrix having evaluated at the least common multiple [xi, xj] of xi and xj as its i, j entry. In this paper, we show for a certain class of arithmetical functions new bounds for det [(xi, xj]), which improve the results obtained by Bourque and Ligh in 1993. As a corollary, we get new lower bounds for det[(xi, xj)], which improve the results obtained by Rajarama Bhat in 1991. We also show for a certain class of semi-multiplicative function new bounds for det([xi, xj]), which improve the results obtained by Bourque and Ligh in 1995.
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Matrices Associated with Classes of Arithmetical Functions
Article
  • Nov 1993
Let S = {x1, x2, ..., xn} be a Set of distinct positive integers. We investigate the structures, determinants, and inverses of n × n matrices of the form [β(xi, xj)], where β is in one of several classes of arithmetical functions. Among the classes we consider are Cohen′s class of even functions (mod r). We also study n × n matrices of the form [f{hook}(xixj)] which have the arithmetical function f{hook}(m) evaluated at the product of xi and xj as their i, j-entry, where f{hook} is a quadratic function.
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Explicit Expressions of Smith's Determinant on a Poset
Article
  • Jan 2001
We introduce the meet-function and join-function on a subset of a poset and use them to give some explicit expressions of Smith's determinant.
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Determinants of Matrices Associated with Incidence Functions on Posets
Article
  • Jun 2004
Let S = x 1,...,x n} be a finite subset of a partially ordered set P. Let f be an incidence function of P. Let [f(xiΛxj)]\left[ {f\left( {x_i \Lambda x_j } \right)} \right] denote the n × n matrix having f evaluated at the meet xiΛxj{x_i \Lambda x_j } of x i and x j as its i, j-entry and [f(xi∨xj)]\left[ {f\left( {x_i \vee x_j } \right)} \right] denote the n × n matrix having f evaluated at the join xi∨xjx_i \vee x_j of x i and x j as its i, j-entry. The set S is said to be meet-closed if [f(xi∨xj)]\left[ {f\left( {x_i \vee x_j } \right)} \right] for all 1 ≤ i, j ≤ n. In this paper we get explicit combinatorial formulas for the determinants of matrices [f(xiΛxj)]\left[ {f\left( {x_i \Lambda x_j } \right)} \right] and [f(xi∨xj)]\left[ {f\left( {x_i \vee x_j } \right)} \right] on any meet-closed set S. We also obtain necessary and sufficient conditions for the matrices [f(xiΛxj)]\left[ {f\left( {x_i \Lambda x_j } \right)} \right] and [f(xi∨xj)]\left[ {f\left( {x_i \vee x_j } \right)} \right] on any meet-closed set S to be nonsingular. Finally, we give some number-theoretic applications.
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On meet matrices on posets
Article
  • Dec 1996
  • LINEAR ALGEBRA APPL
We consider meet matrices on posets as an abstract generalization of greatest common divisor (GCD) matrices. Some of the most important properties of GCD matrices are presented in terms of meet matrices.
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On meet and join matrices associated with incidence functions
Article
  • Oct 2003
  • LINEAR ALGEBRA APPL
We study recently meet matrices on meet-semilattices as an abstract generalization of greatest common divisor (GCD) matrices. Analogously, in this paper we consider join matrices on lattices as an abstract generalization of least common multiple (LCM) matrices. A formula for the determinant of join matrices on join-closed sets, bounds for the determinant of join matrices on all sets and a formula for the inverse of join matrices on join-closed sets are given. The concept of a semi-multiplicative function gives us formulae for meet matrices on join-closed sets and join matrices on meet-closed sets. Finally, we show what new the study of meet and join matrices contributes to the usual GCD and LCM matrices.
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Arithmetical properties of generalized Ramanujan sums
Article
  • May 1972
  • PAC J MATH
The sums studied in this paper are defined as follows. For any two arithmetical functions f and g, let (Formula Present) where the sum extends over the divisors of the greatest common divisor (m, k) of the positive integers m and h. It should be noted that m and k do not enter symmetrically in (1) unless g is constant.
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