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Bounds on the spectrum of nonsingular triangular (0,1)-matrices
Abstract
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+.
... where µ is the Möbius function, was proposed for cn by using a di erent method. Kaarnioja [9] improved the lower bound further and showed that ...
... 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. ...
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?
Let n be a positive integer and S={x1,...,xn} a set of n distinct positive integers. In 1992, Bourque and Ligh showed that if S is factor closed (i.e. d∈S if d|x for any x∈S), then the GCD matrix (S) divides the LCM matrix [S] in the ring of n×n matrices over the integers. In 2002, Hong showed that such factorization holds for any gcd-closed set S (i.e. (xi,xj)∈S for all 1≤i,j≤n) with |S|≤3. But for any integer n≥4, there is a gcd-closed set S with |S|=n such that (S)∤[S]. Meanwhile, Hong proposed the problem of characterizing all gcd-closed sets S with |S|≥4 such that (S)|[S]. For x∈S, let GS(x):={d∈S|d<x,d|xand(d|y|x,y∈S)⇒y∈{d,x}}. In 2009, Feng, Hong and Zhao answered this problem when maxx∈S{|GS(x)|}≤2. Zhao in 2014 and Altinisik, Yildiz and Keskin in 2017 characterized all gcd-closed sets S such that (S)|[S] for the case 4≤|S|≤7 and |S|=8, respectively. In this paper, we introduce a new method to study Hong's problem. We obtain a necessary and sufficient condition on the gcd-closed set S with maxx∈S{|GS(x)|}=3 such that (S)|[S]. This solves partially Hong's problem.
In this paper, improving a famous result of Wolkowicz and Styan for the GCD matrix and the LCM matrix defined on , we present new upper and lower bounds for the smallest and the largest eigenvalues of and in terms of particular arithmetical functions.
In this paper we study the structure and give bounds for the eigenvalues of
the matrix, which ij entry is , 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 .
The eigenvalues and eigenvectors of tridiagonal Toeplitz matrices are known in closed form. This property is in the first part of the paper used to investigate the sensitivity of the spectrum. Explicit expressions for the structured distance to the closest normal matrix, the departure from normality, and the ϵ-pseudospectrum are derived. The second part of the paper discusses applications of the theory to inverse eigenvalue problems, the construction of Chebyshev polynomial-based Krylov subspace bases, and Tikhonov regularization. Copyright © 2012 John Wiley & Sons, Ltd.
Let S={x1,x2,…,xn} be a finite set of distinct positive integers. Throughout this article we assume that the set S is GCD closed. The LCM matrix [S] of the set S is defined to be the n×n matrix with lcm(xi,xj) as its ij element. The famous Bourque-Ligh conjecture used to state that the LCM matrix of a GCD closed set S is always invertible, but currently it is a well-known fact that any nontrivial LCM matrix is indefinite and under the right circumstances it can be even singular (even if the set S is assumed to be GCD closed). However, not much more is known about the inertia of LCM matrices in general. The ultimate goal of this article is to improve this situation. Assuming that S is a meet closed set we define an entirely new lattice-theoretic concept by saying that an element xi∈S generates a double-chain set in S if the set meetcl(CS(xi))∖CS(xi) can be expressed as a union of two disjoint chains (here the set CS(xi) consists of all the elements of the set S that are covered by xi and meetcl(CS(xi)) is the smallest meet closed subset of S that contains the set CS(xi)). We then proceed by studying the values of the Möbius function on sets in which every element generates a double-chain set and use the properties of the Möbius function to explain why the Bourque-Ligh conjecture holds in so many cases and fails in certain very specific instances. After that we turn our attention to the inertia and see that in some cases it is possible to determine the inertia of an LCM matrix simply by looking at the lattice-theoretic structure of (S,|) alone. Finally, we are going to show how to construct LCM matrices in which the majority of the eigenvalues is either negative or positive.
Consider the n×n matrix with (i, j)’th entry gcd (i, j). Its largest eigenvalue λn and sum of entries sn satisfy λn > sn/n. Because sn cannot be expressed algebraically as a function of n, we underestimate it in several ways. In examples, we compare the bounds so obtained with one another and with a bound from S.Hong, R.Loewy (2004). We also conjecture that λn > 6π⁻²nlogn for all n. If n is large enough, this follows from F.Balatoni (1969). © 2016, Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic.
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.
Let be the set of all lower triangular (0,1)-matrices with each diagonal element equal to 1, and let be the minimum of the smallest eigenvalue of as Y goes through . The Ilmonen-Haukkanen-Merikoski conjecture (the IHM conjecture) states that is equal to the smallest eigenvalue of , where with for . In this paper, we present a proof of this conjecture. In our proof we use an inequality for spectral radii of nonnegative matrices.
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 .
In this paper, we present systematically analysis on the smallest eigenvalue of matrices associated with completely even functions (mod r). We obtain several theorems on the asymptotic behavior of the smallest eigenvalue of matrices associated with completely even functions (mod r). In particular, we get information on the asymptotic behavior of the smallest eigenvalue of the famous Smith matrices. Finally some examples are given to demonstrate the main results.
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 , 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.
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.
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)].
Let be an arbitrary strictly increasing infinite sequence of positive integers. For an integer , let . Let be a real number and a given integer. Let \smash{} be the eigenvalues of the power GCD matrix having the power of the greatest common divisor of and as its i,j-entry. We give a nontrivial lower bound depending on and n for \smash{} if . Especially for , this lower bound is given by using the Riemann zeta function. Let be an integer. For a sequence \smash{} satisfying that for any and \smash{}, we show that if , then \smash{}. Let and be any given integers. For the arithmetic progression \smash{}, we show that if , then \smash{}. Finally, we show that for any sequence \smash{} and any \smash{, } approaches infinity when n goes to infinity.
Let be an arbitrary strictly increasing infinite sequence of positive integers. For an integer n≥1, let . Let r>0 be a real number and q≥ 1 a given integer. Let be the eigenvalues of the reciprocal power LCM matrix having the reciprocal power of the least common multiple of xi and xj as its i, j-entry. We show that the sequence converges and . We show that the sequence converges if and . We show also that if r> 1, then the sequence converges and , where t and l are given positive integers such that t≤l−1.
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.
Given an arbitrary strictly increasing infinite sequence {x i} ∞i=1 of positive integers, let S n {x 1,⋯, x n} for any integer n ≥ 1. Let q ≥ 1 be a given integer and f an arithmetical function. Let λ (1)n ≤ ⋯ ≤ λ (n)n be the eigenvalues of the matrix (f(x i, x j)) having f evaluated at the greatest common divisor (x i, x j) of x i and x j as its i, j-entry. We obtain a lower bound depending only on x 1 and n for λ (1)n if (f * μ)(d) ≥ 0 whenever d|x for any x ∈ S n, where g * μ is the Dirichlet convolution of f and μ. Consequently we show that for any sequence {x i} ∞i=1, if f is a multiplicative function satisfying that f(p k) → ∞ as p k → ∞, f(2) > 1 and f(p m) ≥ f(2)f(p m-1) for any prime p and any integer m ≥ 1 and (f (*)μ)(d) > 0 whenever d|x for any x ∈ {x n} ∞n=1, then λ (n-q+1)n approaches infinity when n goes to infinity.