Smith Form Research Articles

Bivariate Polynomial Matrix and Smith Form

Matrix equivalence plays a pivotal role in multidimensional systems, which are typically represented by multivariate polynomial matrices. The Smith form of matrices is one of the important research topics in polynomial matrices. This article mainly investigates the Smith forms of several types of bivariate polynomial matrices and has successfully derived several necessary and sufficient conditions for matrix equivalence.

Quasi-triangularization of matrix polynomials over arbitrary fields

In [19], Taslaman, Tisseur, and Zaballa show that any regular matrix polynomial P(λ) over an algebraically closed field is spectrally equivalent to a triangular matrix polynomial of the same degree. When P(λ) is real and regular, they also show that there is a real quasi-triangular matrix polynomial of the same degree that is spectrally equivalent to P(λ), in which the diagonal blocks are of size at most 2×2. This paper generalizes these results to regular matrix polynomials P(λ) over arbitrary fieldsF, showing that any such P(λ) can be quasi-triangularized to a spectrally equivalent matrix polynomial over F of the same degree, in which the largest diagonal block size is bounded by the highest degree appearing among all of the F-irreducible factors in the Smith form for P(λ).

Equivalence and reduction of bivariate polynomial matrices to their Smith forms

This paper is concerned with Smith forms of bivariate polynomial matrices. For a bivariate polynomial square matrix with the determinant being the product of two distinct and irreducible univariate polynomials, we prove that it is equivalent to its Smith form. We design an algorithm to reduce this class of bivariate polynomial matrices to their Smith forms, and an example is given to illustrate the algorithm. Furthermore, we extend the above class of matrices to a more general case, and derive a necessary and sufficient condition for the equivalence of a matrix and one of its all possible existing Smith forms.

On the recursive equivalence to Smith form of multivariate polynomial matrices

Abstract The Smith form of an $n$-$D$ polynomial matrix plays an important role in many areas of mathematics and engineering. In this paper, we investigate the recursive equivalence problem of $n$-dimensional polynomial matrices, i.e. if diag$(1,B)$ is equivalent to diag$(1,1,C)$, is B equivalent to diag$(1,C)$? We give a negative answer to this question by explicitly constructing a four-dimensional polynomial matrix which is not equivalent to its Smith form.

Smith Form of Triangular Multivariate Polynomial Matrix

Smith Form of Triangular Multivariate Polynomial Matrix

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.

Cyclically presented groups as Labelled Oriented Graph groups

We use results concerning the Smith forms of circulant matrices to identify when cyclically presented groups have free abelianisation and so can be Labelled Oriented Graph (LOG) groups. We generalize a theorem of Odoni and Cremona to show that for a fixed defining word, whose corresponding representer polynomial has an irreducible factor that is not cyclotomic and not equal to ±t, there are at most finitely many n for which the corresponding n-generator cyclically presented group has free abelianisation. We classify when Campbell and Robertson's generalized Fibonacci groups H(r,n,s) are LOG groups and when the Sieradski groups are LOG groups. We prove that amongst Johnson and Mawdesley's groups of Fibonacci type, the only ones that can be LOG groups are Gilbert-Howie groups H(n,m). We conjecture that if a Gilbert-Howie group is a LOG group, then it is a Sieradski group, and prove this in certain cases (in particular, for fixed m, the conjecture can only be false for finitely many n). We obtain necessary conditions for a cyclically presented group to be a connected LOG group in terms of the representer polynomial and apply them to the Prishchepov groups.

Strength analysis of capacitor energy storage cabinet of monorail elevated train

Based on the actual parameters of the capacitor energy storage cabinet on the top of the monorail train, built the cabinet’s finite element model. Then, according to EN 12663-1, set the calibration conditions and fatigue working conditions. Carried out the simulation calculation under different conditions, respectively. The calculation results under the static calibration conditions show that the maximum equivalent stress of each node on the model is smaller than the allowable stress under all working conditions. Therefore, the static strength of the cabinet meets the design requirements. Plotted Goodman fatigue limit diagrams of the cabinet’s base metal and weld and modified them in the Smith form. Then plotted the average stress and stress amplitude under fatigue working conditions in the corresponding scatter diagram. The diagram s show that all points are located within the permitted area. The results show that the fatigue strength of the cabinet meets the requirements of design and use.

Matrices in companion rings, Smith forms, and the homology of 3-dimensional Brieskorn manifolds

We study the Smith forms of matrices of the form f(Cg) where f(t),g(t)∈R[t], where R is an elementary divisor domain and Cg is the companion matrix of the (monic) polynomial g(t). Prominent examples of such matrices are circulant matrices, skew-circulant matrices, and triangular Toeplitz matrices. In particular, we reduce the calculation of the Smith form of the matrix f(Cg) to that of the matrix F(CG), where F,G are quotients of f(t),g(t) by some common divisor. This allows us to express the last non-zero determinantal divisor of f(Cg) as a resultant. A key tool is the observation that a matrix ring generated by Cg – the companion ring of g(t) – is isomorphic to the polynomial ring Qg=R[t]/<g(t)>. We relate several features of the Smith form of f(Cg) to the properties of the polynomial g(t) and the equivalence classes [f(t)]∈Qg. As an application we let f(t) be the Alexander polynomial of a torus knot and g(t)=tn−1, and calculate the Smith form of the circulant matrix f(Cg). By appealing to results concerning cyclic branched covers of knots and cyclically presented groups, this provides the homology of all Brieskorn manifolds M(r,s,n) where r,s are coprime.

Rational forms for block lower-triangular matrices

Linear time invariant (LTI) systems are often represented by rational forms or rational matrix functions. Such forms exhibit important properties of a system, but these properties are often thought to be due to time invariance. Are there such forms for linear time-variant (LTV) systems with comparable properties, generalizing the LTI case? One major obstacle to derive them has been the lack of a divisibility theory for general block lower-triangular matrices, in particular the lack of a Euclidean algorithm or, in its wake, Smith forms and Smith-McMillan forms. But do rational forms for matrices, generalizing the time-invariant properties, exist at all? It turns out they do. The theory presented here produces representations for block lower-triangular matrices as ratios of (block) staircase (or echelon) matrices, and shows how central properties such as co-prime factorization or Bezout identities hold. Instead of the Euclidean algorithm, the theory uses a technique derived from a paper of Paul Van Dooren, namely ‘dead-beat control’. The theory finds good applications in various domains (computational efficiency, determining pre-conditioners, control theory and model reduction via generalized forms of interpolation).

Serre’s Reduction and the Smith Forms of Multivariate Polynomial Matrices

The equivalence of systems plays a critical role in multidimensional systems, which are usually represented by the multivariate polynomial matrices. The Smith form of a matrix is one of the important research contents in polynomial matrices. This paper mainly investigates the Smith forms of some multivariate polynomial matrices. We have obtained several new results and criteria on the reduction of a given multivariate polynomial matrix to its Smith form. These criteria are easily checked by computing the minors of lower order of the given matrix.

The smith form of a multivariate polynomial matrix over an arbitrary coefficient field

ABSTRACT The equivalence of multidimensional systems is closely related to the equivalence of multivariate polynomial matrices, for which the Smith form plays an important role. In this paper we study multivariate polynomial matrices with their entries in the polynomial ring , where is an arbitrary field. We derive some new conditions on reducing these matrices to their Smith forms. These conditions can be verified by computing the reduced Gröbner bases of the associated ideals.

Some results on eigenvalues of finite type, resolvents and Riesz projections

The present paper considers a separable Hilbert space H and a bounded linear operator A:H↦H that has an eigenvalue of finite type at λ0∈C. Using the image and the kernel of certain operators that are defined recursively starting from λ0−A, a construction of the local Smith form and extended canonical system of root functions of λ−A at λ0 is provided. This allows to express the partial multiplicities, the algebraic multiplicity, the Jordan structure and Jordan chains of A at λ0 in terms of the quantities defined by the recursion. Similarly, formulas for the rank, image and kernel of the operators in the principal part of the resolvent (λ−A)−1 at λ0, and thus also of the Riesz projection, are given in terms of the quantities defined by the recursion.

Computing nearby non-trivial Smith forms

We consider the problem of computing the nearest matrix polynomial with a non-trivial Smith Normal Form. We show that computing the Smith form of a matrix polynomial is amenable to numeric computation as an optimization problem. Furthermore, we describe an effective optimization technique to find a nearby matrix polynomial with a non-trivial Smith form. The results are then generalized to include the computation of a matrix polynomial having a maximum specified number of ones in the Smith Form (i.e., with a maximum specified McCoy rank).We discuss the geometry and existence of solutions and how our results can be used for an error analysis. We develop an optimization-based approach and demonstrate an iterative numerical method for computing a nearby matrix polynomial with the desired spectral properties. We also describe an implementation of our algorithms and demonstrate the robustness with examples in Maple.

Root polynomials and their role in the theory of matrix polynomials

We develop a complete and rigorous theory of root polynomials of arbitrary matrix polynomials, i.e., either regular or singular, and study how these vector polynomials are related to the spectral properties of matrix polynomials. We pay particular attention to the so-called maximal sets of root polynomials and prove that they carry complete information about the eigenvalues (finite or infinite) of matrix polynomials and that they are related to the matrices that transform any matrix polynomial into its Smith form. In addition, we describe clearly, for the first time in the literature, the extremality properties of such maximal sets and identify some of them whose vectors have minimal grade. Once the main theoretical properties of root polynomials have been established, the interaction of root polynomials with three problems that have attracted considerable attention in the literature is analyzed. More precisely, we study the change of root polynomials under rational transformations, or reparametrizations, of matrix polynomials, the recovery of the root polynomials of a matrix polynomial from those of its most important linearizations, and the relationship between the root polynomials of two dual pencils. We emphasize that for the case of regular matrix polynomials all the results in this paper can be translated into the language of Jordan chains, as a consequence of the well known relationship between root polynomials and Jordan chains. Therefore, a number of open problems are also solved for Jordan chains of regular matrix polynomials. We also briefly discuss how root polynomials can be used to define eigenvectors and eigenspaces for singular matrix polynomials.

Reduction of Kinetic Equations to Liénard–Levinson–Smith Form: Counting Limit Cycles

We have presented an unified scheme to express a class of system of equations in two variables into a Li\'enard-Levinson-Smith (LLS) oscillator form. We have derived the condition for limit cycle with special reference to Rayleigh and Li\'enard systems for arbitrary polynomial functions of damping and restoring force. Krylov-Boguliubov (K-B) method is implemented to determine the maximum number of limit cycles admissible for a LLS oscillator atleast in the weak damping limit. Scheme is illustrated by a number of model systems with single cycle as well as the multiple cycle cases.

Dual Graded Graphs and Bratteli Diagrams of Towers of Groups

An $r$-dual tower of groups is a nested sequence of finite groups, like the symmetric groups, whose Bratteli diagram forms an $r$-dual graded graph. Miller and Reiner introduced a special case of these towers in order to study the Smith forms of the up and down maps in a differential poset. Agarwal and the author have also used these towers to compute critical groups of representations of groups appearing in the tower. In this paper I prove that when $r=1$ or $r$ is prime, wreath products of a fixed group with the symmetric groups are the only $r$-dual tower of groups, and conjecture that this is the case for general values of $r$. This implies that these wreath products are the only groups for which one can define an analog of the Robinson-Schensted bijection in terms of a growth rule in a dual graded graph.

A New variant of the McEliece cryptosystem based on the Smith form of convolutional codes

ABSTRACTIn this article, the authors propose a new version of the McEliece cryptosystem based on the Smith form of convolutional codes. They use the Smith form to hide a part of the code in the public matrix, and they leave the other part secret. The secret part will then be used for decryption. They hide this part by multiplying it on the left by a random matrix, and they add a random matrix which has a few conditions. Their scheme has a small public key size compared to the original McEliece scheme and resists the unique decoding attack against convolutional structure presented at the conference PQCrypto 2013 by Landais and Tillich. Further, the exhaustive search attack is infeasible on their system.

Automatic Identification of General Vector Error Correction Models

Abstract There are a number of econometrics tools to deal with the different types of situations in which cointegration can appear: I(1), I(2), seasonal, polyno- mial, etc. There are also different kinds of Vector Error Correction models related to these situations. The authors propose a unified theoretical and practical framework to deal with many of these situations. To this aim: (i) they introduce a general class of models and (ii) provide an automatic method to identify models, based on estimating the Smith form of an autoregressive model. Their simulations suggest the power of the new proposed methodology. An empirical example illustrates the methodology.

Inverting a Matrix Function around a Singularity via Local Rank Factorization

This paper proposes a recursive procedure, called the extended local rank factorization (elrf), that characterizes the order of the pole and the coefficients of the Laurent series representation of the inverse of a regular analytic matrix function around a given point. The elrf consists in performing a finite sequence of rank factorizations of matrices of nonincreasing dimension, at most equal to the dimension of the original matrix function. Each step of the sequence is associated with a reduced rank condition, while the termination of the elrf corresponds to a full rank condition; this last step reveals the order of the pole. The Laurent coefficients $B_{n}$ are calculated recursively as $B_{n} = C_{n} + \sum_{k=1}^{n} D_{k} B_{n-k}$, where $C_{n}$, $D_{k}$ have simple closed form expressions in terms of the quantities generated by the elrf. It is also shown that the elrf characterizes the structure of Jordan pairs, Jordan chains, and the local Smith form. The procedure is easily cast in an algorithmic ...