Integer Entries Research Articles

Periodic matrices with integer entries

We give a characterization when a nonzero square matrix with integer entries is periodic using cyclotomic polynomials. We also determine the set of all possible periods of periodic matrices with integer entries via Euler Phi functions.

Rational orthogonal matrices and isomorphism of graphs

Let On(Q) and Sn(Z) be the set of all n by n orthogonal matrices with rational entries and the set of all n by n symmetric matrices with integer entries, respectively. We consider the following question: Given A∈Sn(Z). When does QTAQ∈Sn(Z) with Q∈On(Q) imply that Q is a signed permutation matrix? Wang and Yu [15] show that the answer to the above question is yes whenever the discriminant Δ(A) of the characteristic polynomial of A is odd and square-free. However, the above result cannot be applied for the adjacency matrix of a simple graph, since Δ(A) can never be odd in this situation. We give a new condition under which the answer is still yes for the adjacency matrices of graphs. The result was achieved by employing some tools from our recent work on the generalized spectral characterizations of graphs, which is somewhat unexpected. Some applications of the above result are also provided.

A Riordan group poset

In this paper, we look at Riordan arrays with nonnegative integer entries. In the set, if AT=B for a nonnegative integer matrix T, then we say A≤B. Our primary focus centers around six kinds of key arrays: Appell, Bell, two-Bell, Natural, Hankel, and Pseudo-involution. We examine relations among Hankel, Bell and two-Bell in general, while the other entries are treated case by case. We start by looking at individual cases related to the little and large Schröder numbers. Then we look at the families related to k-Catalan arrays, k-Schröder arrays, and k-Motzkin arrays. We often can give combinatorial interpretations. Along the way, several intriguing counterexamples occur.

Some compactly supported Riesz wavelets associated to any Ed(2)(ℤ) dilation

Let [Formula: see text] be a [Formula: see text] dilation matrix with integer entries and with determinant [Formula: see text]. The existence of Riesz wavelets with compact support associated to [Formula: see text] is proved. Our proof is constructive and the generators of these Riesz wavelets may be taken to be symmetric, with high regularity and many vanishing moments. In our construction, we first study the structure of the quotient group [Formula: see text]. Afterwards and perhaps the main advance in this paper is the fact that we obtain a family of trigonometric polynomials on [Formula: see text] with zeros only in [Formula: see text]. At this point, we are able to get scaling functions with compact support of a multiresolution analysis. In addition, Riesz wavelets with compact support hold by standard multiresolution techniques. Finally, we give some examples of Riesz wavelets where we emphasize on a numerical estimation of their regularity.

On binary matrix properties

The so-called matrix properties of finitely complete categories are a special type of exactness properties that can be encoded as extended matrices of integers. The relation of entailment of matrix properties gives an interesting preorder on the set of such matrices, which can be investigated independently of the category-theoretic considerations. In this paper, we conduct such investigation and obtain several results dealing with binary matrices, i.e., when the only integer entries in the matrix are 0 or 1. Central to these results is a complete description of the preorder in the case of a special type of binary matrices, which we call diagonal matrices, and which include matrices that define Mal’tsev, majority and arithmetical categories.

Irreducibility of Binomials

In this paper, we prove that the family of binomials $x_1^{a_1} \cdots x_m^{a_m}-y_1^{b_1}\cdots y_n^{b_n}$ with $\gcd(a_1, \ldots, a_m, b_1, \ldots, b_n)=1$ is irreducible by identifying the connection between the irreducibility of a binomial in ${\mathbb C}[x_1, \ldots, x_m, y_1, \ldots, y_n]$ and ${\mathbb C}(x_2, \ldots, x_m, y_1, \ldots, y_n)[x_1]$. Then we show that the necessary and sufficient conditions for the irreducibility of this family of binomials is equivalent to the existence of a unimodular matrix $U_i$ with integer entries such that $(a_1, \ldots, a_m, b_1, \ldots, b_n)^T=U_i \be_i$ for $i\in \{1, \ldots, m+n\}$, where $\be_i$ is the standard basis vector.

On a faithful representation of Sturmian morphisms

The set of morphisms mapping any Sturmian sequence to a Sturmian sequence forms together with composition the so-called monoid of Sturm. For this monoid, we define a faithful representation by (3×3)-matrices with integer entries. We find three convex cones in R3 and show that a matrix R∈Sl(Z,3) is a matrix representing a Sturmian morphism if the three cones are invariant under multiplication by R or R−1. This property offers a new tool to study Sturmian sequences. We provide alternative proofs of four known results on Sturmian sequences fixed by a primitive morphism and a new result concerning the square root of a Sturmian sequence.

Farey graph and rational fixed points of the extended modular group

Fixed points of matrices have many applications in various areas of science and mathematics. Extended modular group ¯¯¯¯ΓΓ¯ is the group of 2×22×2 matrices with integer entries and determinant ±1±1. There are strong connections between extended modular group, continued fractions and Farey graph. Farey graph is a graph with vertex set ^Q=Q∪{∞}Q^=Q∪{∞}. In this study, we consider the elements in ¯¯¯¯ΓΓ¯ that fix rationals. For a given rational number, we use its Farey neighbours to obtain the matrix representation of the element in $\overline{\Gamma}$ that fixes the given rational. Then we express such elements as words in terms of generators using the relations between the Farey graph and continued fractions. Finally we give the new block reduced form of these words which all blocks have Fibonacci numbers entries.

On the sum of left and right circulant matrices

We consider square matrices arising as the sum of left and right circulant matrices and derive asymptotics of the sequence of their powers. Particular emphasis is laid on the case where the matrix has consecutive integer entries; we find explicit formulae for the eigenvalues and eigenvectors of the matrix in this case and find its Moore-Penrose pseudoinverse. The calculation involves the discrete Fourier transform of integer vectors arising from sum systems and exhibits a resonance phenomenon.

Optimizing and Factorizing the Wilson Matrix

The Wilson matrix, W, is a 4 × 4 unimodular symmetric positive definite matrix of integers that has been used as a test matrix since the 1940’s, owing to its mild ill-conditioning. We ask how close W is to being the most ill-conditioned matrix in its class, with or without the requirement of positive definiteness. By exploiting the matrix adjugate and applying various matrix norm bounds from the literature we derive bounds on the condition numbers for the two cases and we compare them with the optimal condition numbers found by exhaustive search. We also investigate the existence of factorizations with Z having integer or rational entries. Drawing on recent research that links the existence of these factorizations to number-theoretic considerations of quadratic forms, we show that W has an integer factor Z and two rational factors, up to signed permutations. This little matrix continues to be a useful example on which to apply existing matrix theory as well as being capable of raising challenging questions that lead to new results.

Cationic vacancies as defects in honeycomb lattices with modular symmetries

Layered materials tend to exhibit intriguing crystalline symmetries and topological characteristics based on their two dimensional (2D) geometries and defects. We consider the diffusion dynamics of positively charged ions (cations) localized in honeycomb lattices within layered materials when an external electric field, non-trivial topologies, curvatures and cationic vacancies are present. The unit (primitive) cell of the honeycomb lattice is characterized by two generators, J_1, J_2 in mathrm{SL}_2({mathbb {Z}}) of modular symmetries in the special linear group with integer entries, corresponding to discrete re-scaling and rotations respectively. Moreover, applying a 2D conformal metric in an idealized model, we can consistently treat cationic vacancies as topological defects in an emergent manifold. The framework can be utilized to elucidate the molecular dynamics of the cations in exemplar honeycomb layered frameworks and the role of quantum geometry and topological defects not only in the diffusion process such as prediction of conductance peaks during cationic (de-)intercalation process, but also pseudo-spin and pseudo-magnetic field degrees of freedom on the cationic honeycomb lattice responsible for bilayers.

Further enumeration results concerning a recent equivalence of restricted inversion sequences

Let asc and desc denote respectively the statistics recording the number of ascents or descents in a sequence having non-negative integer entries. In a recent paper by Andrews and Chern, it was shown that the distribution of asc on the inversion sequence avoidance class $I_n(\geq,\neq,>)$ is the same as that of $n-1-\text{asc}$ on the class $I_n(>,\neq,\geq)$, which confirmed an earlier conjecture of Lin. In this paper, we consider some further enumerative aspects related to this equivalence and, as a consequence, provide an alternative proof of the conjecture. In particular, we find recurrence relations for the joint distribution on $I_n(\geq,\neq,>)$ of asc and desc along with two other parameters, and do the same for $n-1-\text{asc}$ and desc on $I_n(>,\neq,\geq)$. By employing a functional equation approach together with the kernel method, we are able to compute explicitly the generating function for both of the aforementioned joint distributions, which extends (and provides a new proof of) the recent result $|I_n(\geq,\neq,>)|=|I_n(>,\neq,\geq)|$. In both cases, an algorithm is formulated for computing the generating function of the asc distribution on members of each respective class having a fixed number of descents.

LU and CR Elimination

The reduced row echelon form ${\bf rref}(A)$ has traditionally been used for classroom examples: small matrices $A$ with integer entries and low rank $r$. This paper creates a column-row rank-revealing factorization ${A=CR}$, with the first $r$ independent columns of $A$ in $C$ and the $r$ nonzero rows of ${\bf rref}(A)$ in $R$. We want to reimagine the start of a linear algebra course by helping students to see the independent columns of $A$ and the rank and the column space. If $B$ contains the first $r$ independent rows of $A$, then those rows of ${A=CR}$ produce ${B=WR}$. The $r$ by $r$ matrix $W$ has full rank $r$, where $B$ meets $C$. Then the triple factorization ${A=CW^{-1}B}$ treats columns and rows of $A$ ($C$ and $B$) in the same way.

There is no Diophantine D(−1)$D(-1)$‐quadruple

A set of positive integers with the property that the product of any two of them is the successor of a perfect square is called Diophantine D ( − 1 ) $D(-1)$ -set. Such objects are usually studied via a system of generalized Pell equations naturally attached to the set under scrutiny. In this paper, an innovative technique is introduced in the study of Diophantine D ( − 1 ) $D(-1)$ -quadruples. The main novelty is the uncovering of a quadratic equation relating various parameters describing a hypothetical D ( − 1 ) $D(-1)$ -quadruple with integer entries. In combination with extensive computations, this idea leads to the confirmation of the conjecture according to which there is no Diophantine D ( − 1 ) $D(-1)$ -quadruples.

On $m$-th roots of nilpotent matrices

A new necessary and sufficient condition for the existence of an $m$-th root of a nilpotent matrix in terms of the multiplicities of Jordan blocks is obtained and expressed as a system of linear equations with nonnegative integer entries which is suitable for computer programming. Thus, computation of the Jordan form of the $m$-th power of a nilpotent matrix is reduced to a single matrix multiplication; conversely, the existence of an $m$-th root of a nilpotent matrix is reduced to the existence of a nonnegative integer solution to the corresponding system of linear equations. Further, an erroneous result in the literature on the total number of Jordan blocks of a nilpotent matrix having an $m$-th root is corrected and generalized. Moreover, for a singular matrix having an $m$-th root with a pair of nilpotent Jordan blocks of sizes $s$ and $l$, a new $m$-th root is constructed by replacing that pair by another one of sizes $s+i$ and $l-i$, for special $s,l,i$. This method applies to solutions of a system of linear equations having a special matrix of coefficients. In addition, for a matrix $A$ over an arbitrary field that is a sum of two commuting matrices, several results for the existence of $m$-th roots of $A^k$ are obtained.

Equivalent polyadic decompositions of matrix multiplication tensors

Invariance transformations of polyadic decompositions of matrix multiplication tensors define an equivalence relation on the set of such decompositions. In this paper, we present an algorithm to efficiently decide whether two polyadic decompositions of a given matrix multiplication tensor are equivalent. With this algorithm, we analyze the equivalence classes of decompositions of several matrix multiplication tensors. This analysis is relevant for the study of fast matrix multiplication as it relates to the question of how many essentially different fast matrix multiplication algorithms there exist. This question has been first studied by de Groote, who showed that for the multiplication of 2 ×2 matrices with 7 active multiplications, all algorithms are essentially equivalent to Strassen’s algorithm. In contrast, the results of our analysis show that for the multiplication of larger matrices (e.g., 2 ×3 by 3 ×2 or 3 ×3 by 3 ×3 matrices), two decompositions are very likely to be essentially different. We further provide a necessary criterion for a polyadic decomposition to be equivalent to a polyadic decomposition with integer entries. Decompositions with specific integer entries, e.g., powers of two, provide fast matrix multiplication algorithms with better efficiency and stability properties. This condition can be tested algorithmically and we present the conclusions obtained for the decompositions of small/medium matrix multiplication tensors.

On positional representation of integer vectors

We show that any m×m matrix M with integer entries and det⁡M=Δ≠0 can be equipped by a finite digit set D⊂Zm such that any integer m-dimensional vector belongs to the setFinD(M)={∑k∈IMkdk:∅≠I finite subset of Z and dk∈D for each k∈I}⊂⋃k∈N1ΔkZm. We also characterize the matrices M for which the sets FinD(M) and ⋃k∈N1ΔkZm coincide.

The finiteness conjecture holds in * *The first author is partially supported by the MIUR Grant E83C18000100006 Regular and stochastic behaviour in dynamical systems.

Let A, B be matrices in having trace greater than or equal to 2. Assume the pair A, B is coherently oriented, that is, can be conjugated to a pair having nonnegative entries. Assume also that either A, B −1 is coherently oriented as well, or A, B have integer entries. Then the Lagarias–Wang finiteness conjecture holds for the set {A, B}, with optimal product in {A, B, AB, A 2 B, AB 2}. In particular, it holds for every pair of 2 × 2 matrices with nonnegative integer entries and determinant 1.

Integer matrix factorisations, superalgebras and the quadratic form obstruction

We identify and analyse obstructions to factorisation of integer matrices into products NTN or N2 of matrices with rational or integer entries. The obstructions arise as quadratic forms with integer coefficients and raise the question of the discrete range of such forms. They are obtained by considering matrix decompositions over a superalgebra. We further obtain a formula for the determinant of a square matrix in terms of adjugates of these matrix decompositions, as well as identifying a co-Latin symmetry space.

Permanents of 2 x 2 Matrices Modulo n

In this article we compute the number of invertible two by two matrices with integer entries modulo n whose permanents are congruent modulo n to a given integer x.