Smith Normal Form Research Articles

An extended GCRD algorithm for parametric univariate polynomial matrices and application to parametric Smith form

The first extended greatest common right divisor (GCRD) algorithm for parametric univariate polynomial matrices is presented. The starting point of this GCRD algorithm is the free property of submodules over univariate polynomial rings. We convert the computation of GCRDs to that of free basis for modules and prove that a free basis of the submodule generated by row vectors of input matrices forms just a GCRD of these matrices. The GCRD algorithm is obtained by computing a minimal Gröbner basis for the corresponding submodule since a minimal Gröbner basis of submodules is a free basis for univariate cases. While the key idea of extended algorithm is to construct a special module by adding the unit vectors which can record the representation coefficients. This method based on modules can be naturally generalized to the parametric case because of the comprehensive Gröbner systems for modules. As a consequence, we obtain an extended GCRD algorithm for parametric univariate polynomial matrices. More importantly, we apply the proposed extended GCD algorithm for univariate polynomials (as a special case of matrices) to the computation of Smith normal form, and give the first algorithm for reducing a univariate polynomial matrix with parameters to its Smith normal form.

Codeterminantal graphs

We introduce the concept of codeterminantal graphs, which generalize the concepts of cospectral and coinvariant graphs. To do this, we investigate the relationship of the spectrum and the Smith normal form (SNF) with the determinantal ideals. We establish a necessary and sufficient condition for graphs to be codeterminantal on R[x], and we present some computational results on codeterminantal graphs up to 9 vertices. Finally, we show that complete graphs and star graphs are determined by the SNF of its distance Laplacian matrix.

A Formalization of the Smith Normal Form in Higher-Order Logic

This work presents formal correctness proofs in Isabelle/HOL of algorithms to transform a matrix into Smith normal form, a canonical matrix form, in a general setting: the algorithms are written in an abstract form and parameterized by very few simple operations. We formally show their soundness provided the operations exist and satisfy some conditions, which always hold on Euclidean domains. We also provide a formal proof on some results about the generality of such algorithms as well as the uniqueness of the Smith normal form. Since Isabelle/HOL does not feature dependent types, the development is carried out by switching conveniently between two different existing libraries by means of the lifting and transfer package and the use of local type definitions, a sound extension to HOL.

Rational matrix digit systems

Let A be a matrix with rational entries which has no eigenvalue of absolute value and let be the smallest nontrivial A-invariant -module. We lay down a theoretical framework for the construction of digit systems , where finite, that admit finite expansions of the form for every element . We put special emphasis on the explicit computation of small digit sets that admit this property for a given matrix A, using techniques from matrix theory, convex geometry, and the Smith Normal Form. Moreover, we provide a new proof of general results on this finiteness property and recover analogous finiteness results for digit systems in number fields a unified way.

On Smith normal forms of q-Varchenko matrices

In this paper, we investigate q-Varchenko matrices for some hyperplane arrangements with symmetry in two andthree dimensions, and prove that they have a Smith normal formover Z[q]. In particular, we examine the hyperplane arrangement forthe regular n-gon in the plane and the dihedral model in the spaceand Platonic polyhedra. In each case, we prove that the q-Varchenko matrix associated with the hyperplane arrangement has a Smith normal form over Z[q] and realize their congruent transformation matrices over Z[q] as well.

Triangular form of Laurent polynomial matrices and their factorization

The issue of the semiscalar equivalence of Laurent polynomial matrices is investigated and the triangular form of such matrices and their finite sets is established with respect to this equivalence. The theorem on regularization of a Laurent polynomial matrix is proved. This theorem is used in the problem of factorization of such matrices. The factorization criterion of a Laurent polynomial matrix with a regular multiplier with a predetermined Smith normal form is obtained.

Limits and fluctuations of p-adic random matrix products

We show that singular numbers (also known as elementary divisors, invariant factors or Smith normal forms) of products and corners of random matrices over $${\mathbb {Q}}_p$$ are governed by the Hall–Littlewood polynomials, in a structurally identical manner to the known relations between singular values of complex random matrices and Heckman–Opdam hypergeometric functions. This implies that the singular numbers of a product of corners of Haar-distributed elements of $$\mathrm {GL}_N({\mathbb {Z}}_p)$$ form a discrete-time Markov chain distributed as a Hall–Littlewood process, with the number of matrices in the product playing the role of time. We give an exact sampling algorithm for the Hall–Littlewood processes which arise by relating them to an interacting particle system similar to PushTASEP. By analyzing the asymptotic behavior of this particle system, we show that the singular numbers of such products obey a law of large numbers and their fluctuations converge dynamically to independent Brownian motions. In the limit of large matrix size, we also show that the analogues of the Lyapunov exponents for matrix products have universal limits within this class of $$\mathrm {GL}_N({\mathbb {Z}}_p)$$ corners.

The ultradiscrete Toda lattice and the Smith normal form of bidiagonal matrices

The discrete Toda lattice preserves the eigenvalues of tridiagonal matrices, and convergence of dependent variables to the eigenvalues can be proved under appropriate conditions. We show that the ultradiscrete Toda lattice preserves invariant factors of a certain bidiagonal matrix over a principal ideal domain and prove convergence of dependent variables to invariant factors using properties of the box and ball system. Using this fact, we present a new method for computing the Smith normal form of a given matrix.

Unique sums and differences in finite Abelian groups

Let A,B be subsets of a finite abelian group G. Suppose that A+B does not contain a unique sum, i.e., there is no g∈G with a unique representation g=a+b, a∈A, b∈B. From such sets A,B, sparse linear systems over the rational numbers arise. We obtain a new determinant bound on invertible submatrices of the coefficient matrices of these linear systems. Under the condition that |A|+|B| is small compared to the order of G, these bounds provide essential information on the Smith Normal Form of these coefficient matrices. We use this information to prove that A and B admit coset partitions whose parts have properties resembling those of A and B. As a consequence, we improve previously known sufficient conditions for the existence of unique sums in A+B and show how our structural results can be used to classify sets A and B for which A+B does not contain a unique sum when |A|+|B| is relatively small. Our method also can be applied to subsets of abelian groups which have no unique differences.

On the Smith normal form of least common multiple of matrices from some class of matrices

On the Smith normal form of least common multiple of matrices from some class of matrices

On lattice point counting in $$\varDelta $$-modular polyhedra

Let a polyhedron $P$ be defined by one of the following ways: (i) $P = \{x \in R^n \colon A x \leq b\}$, where $A \in Z^{(n+k) \times n}$, $b \in Z^{(n+k)}$ and $rank\, A = n$; (ii) $P = \{x \in R_+^n \colon A x = b\}$, where $A \in Z^{k \times n}$, $b \in Z^{k}$ and $rank\, A = k$. And let all rank order minors of $A$ be bounded by $\Delta$ in absolute values. We show that the short rational generating function for the power series $$ \sum\limits_{m \in P \cap Z^n} x^m $$ can be computed with the arithmetic complexity $ O\left(T_{SNF}(d) \cdot d^{k} \cdot d^{\log_2 \Delta}\right), $ where $k$ and $\Delta$ are fixed, $d = \dim P$, and $T_{SNF}(m)$ is the complexity to compute the Smith Normal Form for $m \times m$ integer matrix. In particular, $d = n$ for the case (i) and $d = n-k$ for the case (ii). The simplest examples of polyhedra that meet conditions (i) or (ii) are the simplicies, the subset sum polytope and the knapsack or multidimensional knapsack polytopes. We apply these results to parametric polytopes, and show that the step polynomial representation of the function $c_P(y) = |P_{y} \cap Z^n|$, where $P_{y}$ is parametric polytope, can be computed by a polynomial time even in varying dimension if $P_{y}$ has a close structure to the cases (i) or (ii). As another consequence, we show that the coefficients $e_i(P,m)$ of the Ehrhart quasi-polynomial $$ \left| mP \cap Z^n\right| = \sum\limits_{j = 0}^n e_i(P,m)m^j $$ can be computed by a polynomial time algorithm for fixed $k$ and $\Delta$.

Enumeration of cospectral and coinvariant graphs

We present enumeration results on the number of connected graphs up to 10 vertices for which there is at least one other graph with the same spectrum (cospectral mate), or at least one other graph with the same Smith normal form (coinvariant mate) with respect to several matrices associated to a graph. The presented numerical data give some indication that possibly the Smith normal form of the distance Laplacian and the signless distance Laplacian matrices could be a finer invariant than the spectrum to distinguish graphs. Finally, we prove a graph characterization using the Smith normal form of the distance signless Laplacian matrix.

Filling anomaly for general two- and three-dimensional C4 symmetric lattices

In this manuscript, we derive symmetry indicator formulas for the filling anomaly on 2D square lattices with and without time reversal, inversion symmetry, or their product, in the presence of spin-orbit coupling. We go beyond previous work by considering lattices with atoms occupying multiple Wyckoff positions. We also provide an algorithm using the Smith normal form that systematizes the derivation. The formulas determine the corner charge in 2D atomic or fragile topological insulators, as well as in 3D insulators and semimetals by studying their 2D slices. We apply our results to a 3D tight-binding model on a body-centered tetragonal lattice, whose projection into the 2D plane has two atoms in the unit cell. Our symmetry indicators correctly describe the higher-order hinge states and Fermi arcs in cases where the existing indicators do not apply.

矩阵多项式的Bezout等式

本文研究经典的多项式理论中的Bezout等式在矩阵多项式理论中的情形。使用矩阵多项式的Smith标准型证明了一个关于Bezout等式的推广定理。 This work studies the Bezout equation in matrix polynomials. A theorem generalized the Bezout theorem is proved by the Smith normal form of a matrix polynomial.

On the Smith normal form of walk matrices

Let G be a graph with n vertices. The walk matrix W(G) of G is the matrix [e,Ae,…,An−1e], where A is the adjacency matrix of G and e is the all-one vector. Let W be a walk matrix of order n. We show that at most ⌊n2⌋ invariant factors of W are congruent to 2 modulo 4. As a consequence, it is proved that, for any n×n walk matrix W with 2-rank r, the determinant of W is always a multiple of 2⌈3n−4r2⌉. Moreover, if 2−⌈3n−4r2⌉det⁡W is odd and square-free, then the Smith normal form of W can be recovered uniquely from the triple (n,r,det⁡W).

On the Smith normal form of Varchenko matrix

Let A be a hyperplane arrangement in A isomorphic to ℝ n . Let V q be the q -Varchenko matrix for the arrangement A with all hyperplane parameters equal to q . In this paper, we consider three interesting cases of q -Varchenko matrices associated to hyperplane arrangements. We show that they have a Smith normal form over ℤ[ q ] .

On solvability of the matrix equation AXB = C over a principal ideal domain

In this paper we present conditions of solvability of the matrix equation AXB = B over a principal ideal domain. The necessary and sufficient conditions of solvability of equation AXB = B in term of the Smith normal forms and in term of the Hermi-te normal forms of matrices constructed in a certain way by using the coefficients of this equation are proposed. If a solution of this equation exists we propose the method for its construction.

Smith Normal Form of a distance matrix inspired by the four-point condition

The four point theorem is a condition for distances to arise from trees. Based on this condition, for any tree T on n vertices, we associate an (n2)×(n2) matrix MT.We find the rank and the Smith Normal Form (SNF) of the matrix MT and show that it only depends on n and is independent of the structure of the tree T. Curiously, the non-zero part of the SNF of MT coincides with the SNF of the distance matrix of T. Many such “tree independent” results are known and this result is yet another such result.

The Smith normal form distribution of a random integer matrix

We show that the density μ of the Smith normal form (SNF) of a random integer matrix exists and equals a product of densities μps of SNF over Z/psZ with p a prime and s some positive integer. Our approach is to connect the SNF of a matrix with the greatest common divisors (gcds) of certain polynomials of matrix entries, and develop the theory of multi-gcd distribution of polynomial values at a random integer vector. We also derive a formula for μps and determine the density μ for several interesting types of sets.

On Serre Reduction of Multidimensional Systems

Serre reduction of a system plays a key role in the theory of Multidimensional systems, which has a close connection with Serre reduction of polynomial matrices. In this paper, we investigate the Serre reduction problem for two kinds of nD polynomial matrices. Some new necessary and sufficient conditions about reducing these matrices to their Smith normal forms are obtained. These conditions can be easily checked by existing Gröbner basis algorithms of polynomial ideals.