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On meet hypermatrices and their eigenvalues

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Abstract

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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... Recently, Ilmonen [11] discovered an explicit polyadic decomposition of meet tensors when the domain of definition is closed under the meet operation and generalized previously known bounds on the eigenvalues of meet matrices to formulate new bounds on the eigenvalues of meet tensors. The structure-theoretic result can be interpreted as a generalization of the LDL T decompositions derived for meet and join tensors in the existing literature [12,13]. ...
... This approach covers, e.g., the aforementioned class of LCM tensors, which are a special case of join tensors defined on the divisor lattice. The approach taken in this paper generalizes the results of [11], provides a constructive version of the result (1) for join tensors defined on general join semilattices, and gives additional insight on the structure of join tensors. In this paper, the focus is kept on join tensors since it covers the important special case of LCM tensors in a natural way, but the results are straightforward to generalize to meet tensors as well. ...
... In particular, numerical schemes based on the power iteration [17] and shifted power iteration [18] have been developed for the solution of extremal tensor eigenvalues. As an application of the decompositions developed in this paper, we consider tensor eigenvalues of join tensors and state a generalization of the bound discovered by Ilmonen [11] in the framework of general join tensors. We assess the sharpness of this upper bound for a class of LCM tensors in an ensemble of test cases utilizing the explicit tensor-train decomposition in the numerical solution of the dominant eigenvalues of LCM tensors. ...
Article
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.
... The focus of the latticetheoretic community is currently shifting to tensors as well: Haukkanen [18] considered the (Cayley) hyperdeterminants of GCD tensors and Luque [31] studied (Cayley) hyperdeterminants of meet tensors. Ilmonen [21] considered eigenvalues of meet tensors, and the following result was proven. |f (x k ∧ x i2 ∧ · · · ∧ x i d )|. ...
... , x n } is a finite, meet closed subset of P ordered x i x j only if i ≤ j, and that f is a complex-valued function on P . It has been shown in [21] that the meet tensor (S d ) f has an explicit polyadic decomposition given by ...
... By setting y = x in (5.1), it can also be verified that the Smith tensor A is positive definite since p(x) = Ax d > 0 for all x ∈ R n \ {0}. Conditions for the positive definiteness of general meet tensors have been discussed in [21]. An ensemble of Smith tensors was generated using the sparse TT-decomposition (4.2) and the H-and Z-eigenvalues were computed using (i) Moderate even order between 2 ≤ d ≤ 20 ( Figure 3). ...
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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.
... This article has obtained some new promotion to the gcd-matrix based on the key outcomes in [17, 24,25]. These properties are widely used in the communication theory, the algebraic coding theory, the cryptography and other fields [3,9,15]. For the convenience of introduction, the following definitions have been given first. ...
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Preface.- Introduction.- General Discriminants and Resultants.- Projective Dual Varieties and General Discriminants.- The Cayley Method of Studying Discriminants.- Associated Varieties and General Resultants.- Chow Varieties.- Toric Varieties.- Newton Polytopes and Chow Polytopes.- Triangulations and Secondary Polytopes.- A-Resultants and Chow Polytopes of Toric Varieties.- A-Discriminants.- Principal A-Discriminants.- Regular A-Determinants and A-Discriminants.- Classical Discriminants and Resultants.- Discriminants and Resultants for Polynomials in One Variable.- Discriminants and Resultants for Forms in Several Variables.- Hyperdeterminants.- Appendix A. Determinants.- Appendix B. A. Cayley: On the Theory of Elimination.- Bibliography.- Notes and References.- List of Notation.- Index
Book
Preface. Basic Symbols. Basic Notations. I. Euler's phi-function. II. The arithmetical function d(n), its generalizations and its analogues. III. Sum-of-divisors function, generalizations, analogues Perfect numbers and related problems. IV. P, p, B, beta and related functions. V. omega(n), Omega(n) and related functions. VI. Function mu k-free and k-full numbers. VII. Functions pi(x), psi(x), theta(x), and the sequence of prime numbers. VIII. Primes in arithmetic progressions and other sequences. IX. Additive and diophantine problems involving primes. X. Exponential sums. XI. Character sums. XII. Binomial coefficients, consecutive integers and related problems. XIII. Estimates involving finite groups and semi-simple rings. XIV. Partitions. XV. Congruences, residues and primitive roots. XVI. Additive and multiplicative functions. Index of authors.
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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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In this paper, we define the symmetric hyperdeterminant, eigenvalues and E-eigenvalues of a real supersymmetric tensor. We show that eigenvalues are roots of a one-dimensional polynomial, and when the order of the tensor is even, E-eigenvalues are roots of another one-dimensional polynomial. These two one-dimensional polynomials are associated with the symmetric hyperdeterminant. We call them the characteristic polynomial and the E-characteristic polynomial of that supersymmetric tensor. Real eigenvalues (E-eigenvalues) with real eigenvectors (E-eigenvectors) are called H-eigenvalues (Z-eigenvalues). When the order of the supersymmetric tensor is even, H-eigenvalues (Z-eigenvalues) exist and the supersymmetric tensor is positive definite if and only if all of its H-eigenvalues (Z-eigenvalues) are positive. An -order n-dimensional supersymmetric tensor where m is even has exactly n(m−1)n−1 eigenvalues, and the number of its E-eigenvalues is strictly less than n(m−1)n−1 when m≥4. We show that the product of all the eigenvalues is equal to the value of the symmetric hyperdeterminant, while the sum of all the eigenvalues is equal to the sum of the diagonal elements of that supersymmetric tensor, multiplied by (m−1)n−1. The n(m−1)n−1 eigenvalues are distributed in n disks in . The centers and radii of these n disks are the diagonal elements, and the sums of the absolute values of the corresponding off-diagonal elements, of that supersymmetric tensor. On the other hand, E-eigenvalues are invariant under orthogonal transformations.
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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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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.
Conference Paper
We propose a theory of eigenvalues, eigenvectors, singular values, and singular vectors for tensors based on a constrained variational approach much like the Rayleigh quotient for symmetric matrix eigenvalues. These notions are particularly useful in generalizing certain areas where the spectral theory of matrices has traditionally played an important role. For illustration, we will discuss a multilinear generalization of the Perron-Frobenius theorem
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We compute hyperdeterminants of hypermatrices whose indices belongs in a meet-semilattice and whose entries depend only of the greatest lower bound of the indices. One shows that an elementary expansion of such a polynomial allows to generalize a theorem of Lindstr\"om to higher-dimensional determinants. And we gave as an application generalizations of some results due to Lehmer, Li and Haukkanen.
Corrected reprint of the 1986 original Cambridge studies in advanced mathematics
  • Rp Stanley
Stanley RP. Enumerative combinatorics. Vol. 1. Corrected reprint of the 1986 original. Vol. 49, Cambridge studies in advanced mathematics. Cambridge: Cambridge University Press; 1997.
Hyperdeterminants on semilattices. Linear Multilinear Algebra
  • J-G Luque
Luque J-G. Hyperdeterminants on semilattices. Linear Multilinear Algebra. 2008;56:333-344.
Corrected reprint of the 1986 original
  • R P Stanley
Stanley RP. Enumerative combinatorics. Vol. 1. Corrected reprint of the 1986 original. Vol. 49, Cambridge studies in advanced mathematics. Cambridge: Cambridge University Press; 1997.