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Definirali bomo relacijo leve ekvivalence na matrikah, katerih elementi pripadajo nekemu glavnemu kolobarju. Poiskali bomo Hermitovo normalno formo in obravnavali njene posledice. We will define the left equivalence relation on matrices whose elements belong to a principal ideal ring. We will find the Hermite normal form and explore some of its consequences.

A coupling extended multiscale finite element method (P-CEMsFEM) is developed for the numerical analysis of thermoelastic problems with polygonal microstructures. In this method, the polygonal microstructures are effectively represented by polygonal coarse-grid elements and the corresponding numerical base functions are constructed for the temperature and displacement fields, respectively, by a unified method with the corresponding equivalent matrices. To reflect the interaction of deformations among different directions, the additional coupling terms are introduced into the numerical base functions. In addition, an improved downscaling technique is proposed to directly obtain the satisfying microscopic solutions in the P-CEMsFEM. Moreover, an arbitrary multi-node strategy is developed to further improve the computational accuracy for the two-dimensional thermoelastic problems. Two types of representative numerical examples are presented. The first type examples are given to testify the proposed multiscale method and the results indicate that the P-CEMsFEM has high accuracy and efficiency for the thermoelastic analysis of heterogeneous multiphase materials and structures. The second type examples testify that the P-CEMsFEM is applicable for practical engineering problems.

This paper is concerned with the factorization and equivalence problems of multivariate polynomial matrices. We present a new criterion for the existence of matrix factorizations for a class of multivariate polynomial matrices, and prove that these matrix factorizations are unique. Based on this new criterion and the constructive proof process, we give an algorithm to compute a matrix factorization of a multivariate polynomial matrix. After that, we put forward a sufficient and necessary condition for the equivalence of square polynomial matrices: a square polynomial matrix is equivalent to a diagonal triangle if it satisfies the condition. An illustrative example is given to show the effectiveness of the matrix equivalence theorem.

A new matrix product, called the semi-tensor product (STP), is briefly reviewed. The STP extends the classical matrix product to two arbitrary matrices. Under STP the set of matrices becomes a monoid (semi-group with identity). Some related structures and properties are investigated. Then the generalized matrix addition is also introduced, which extends the classical matrix addition to a class of two matrices with different dimensions. Motivated by STP of matrices, two kinds of equivalences of matrices (including vectors) are introduced, which are called matrix equivalence (M-equivalence) and vector equivalence (V-equivalence) respectively. The lattice structure has been established for each equivalence. Under each equivalence, the corresponding quotient space becomes a vector space. Under M-equivalence, many algebraic, geometric, and analytic structures have been posed to the quotient space, which include (i) lattice structure; (ii) inner product and norm (distance); (iii) topology; (iv) a fiber bundle structure, called the discrete bundle; (v) bundled differential manifold; (vi) bundled Lie group and Lie algebra. Under V-equivalence, vectors of different dimensions form a vector space ${\cal V}$, and a matrix $A$ of arbitrary dimension is considered as an operator (linear mapping) on ${\cal V}$. When $A$ is a bounded operator (not necessarily square but includes square matrices as a special case), the generalized characteristic function, eigenvalue and eigenvector etc. are defined. In one word, this new matrix theory overcomes the dimensional barrier in certain sense. It provides much more freedom for using matrix approach to practical problems.

We obtain a canonical form for the left ( 2 , 1 ) -Wiener–Hopf equivalence at infinity for polynomial matrices of order 3. This equivalence relation appears in a problem of structured perturbation when we consider polynomial matrix representations associated with controllable pairs.

This paper attempts to prove the D-optimality of the saturated designs X ∗ and X ∗ ∗ of order 22, already existing in the current literature. The corresponding non-equivalent information matrices M ∗ =(X ∗ ) T X ∗ and M ∗ ∗ =(X ∗ ∗ ) T X ∗ ∗ have the maximum determinant. Within the application of a specific procedure, all symmetric and positive definite matrices M of order 22 with determinant the square of an integer and ≥ det(M ∗ ) are constructed. This procedure has indicated that there are 26 such non-equivalent matrices M, for 24 of which the non-existence of designs X such that X T X =M is proved. The remaining two matrices M are the information matrices M ∗ and M ∗ ∗ .

A novel equivalence principle algorithm (EPA) domain decomposition method previously for the surface integral equation is explored for volume integral equation in this work. The whole object is decomposed into several subdomains, the equivalence spheres are automatically produced to enclose each subdomain. The Rao–Wilton–Glisson (RWG) and body of revolution (BoR) basis functions are defined on the equivalence sphere surface. The equivalence currents defined on the RWG basis functions are evaluated with multilevel fast multipole algorithm, and then the currents defined on the RWGs can be projected onto the BoR basis functions. Compared with surface integral equation EPA-BoR algorithm, the ratio of the number of average original unknowns in each subdomain with respect to the equivalence unknowns is relatively larger, which leads to significant reduction of computations. Numerical simulations validate the accuracy and efficiency of the proposed method.

In this work, we consider the properties of fractal correlation of fluctuations of primary differential polarization parameters. The spatial correlation function for isotropic light scattering is first generalized to light scattering by the anisotropic medium. It is then demonstrated that when there exist finite fractal correlations between fluctuations of the elementary polarization properties the elements of the differential Mueller matrix are functions of the thickness of the layer of the medium. Thus the differential Mueller matrices derived from the logarithm of measured macroscopic Mueller matrices can only be regarded as equivalent matrices. The results obtained in this work are helpful for correctly interpreting measured macroscopic Mueller matrices.

We establish the conditions of semiscalar equivalence for one class of regularizable polynomial matrices and determine the canonical form of the matrix relative to the quasidiagonal similarity.

We consider the problem of determining whether two polynomial matrices can be transformed to one another by left multiplying with some nonsingular numerical matrix and right multiplying by some invertible polynomial matrix. Thus the equivalence relation arises. This equivalence relation is known as semiscalar equivalence. Large difficulties in this problem arise already for 2-by-2 matrices. In this paper the semiscalar equivalence of polynomial matrices of second order is investigated. In particular, necessary and sufficient conditions are found for two matrices of second order being semiscalarly equivalent. The main result is stated in terms of determinants of Toeplitz matrices.

It is known that the stability of a Metzler matrix can be characterized in a Routh–Hurwitz-like fashion based on a recursive application of scalar Schur complements [1] . Our objective in this brief note is to show that recently obtained stability conditions are equivalent statements of this result and can be deduced directly therefrom using only elementary results from linear algebra. Implications of this equivalence are also discussed and several examples are given to illustrate potentially interesting system-theoretic applications of this observation.

In this paper, we study invariant control systems that generalise positive systems. A characterisation of linear control systems invariant on polyhedral cones (corner regions) in the state-space, called cone-invariant linear control systems, is established both for the inputs taking values in a polyhedral cone in the control space and for the inputs taking values in an affine polyhedral cone. The problem of equivalence between control systems invariant on corner regions is introduced. For cone-invariant linear control systems, we study invariance-preserving state-equivalence and invariance-preserving feedback-equivalence and present characterisations of both notions of equivalence.

The elementary operations on matrices are of great importance and utility since they are applied in obtaining the so-called equivalent matrices, which are related to each other in many important attributes. In this paper we describe a new package for dealing with the elementary matrix operations of a given matrix. The package, developed by the authors in the freeware program Maxima, version 5.43.2, incorporates two commands which have a fairly intuitive syntax and allow the user indicating one or more elementary operations together so that they are applied to the rows (or columns) of a given matrix. Such features make it more effective and friendly than the default built-in commands in Maxima to perform such operations. In addition, our outputs are consistent with Maxima’s notation. Several illustrative examples, aimed to show the good performance of the package, are also given.

We introduce the notion of balanced strong shift equivalence between square nonnegative integer matrices, and show that two finite graphs with no sinks are one-sided eventually conjugate if and only if their adjacency matrices are conjugate to balanced strong shift equivalent matrices. Moreover, we show that such graphs are eventually conjugate if and only if one can be reached by the other via a sequence of out-splits and balanced in-splits; the latter move being a variation of the classical in-split move introduced by Williams in his study of shifts of finite type. We also relate one-sided eventual conjugacies to certain block maps on the finite paths of the graphs. These characterizations emphasize that eventual conjugacy is the one-sided analog of two-sided conjugacy., Minor changes, examples have been distributed throughout the manuscript, 21 pages. This is the published version

Within the algebraic analysis approach to linear system theory, a multidimensional linear system can be studied by means of its associated finitely presented left module. Deep connections exist between module isomorphisms and equivalent matrices. In the present paper, we introduce a constructive proof of a result due to Warfield which controls the size of equivalent matrices involved in the study of isomorphic modules. We illustrate our constructive proof with an example coming from differential equations with constant coefficients.

In the present article, the authors continue the line of inquiry started by Cigler and Jerman, who studied the separation of eigenvalues of a matrix under an action of a matrix group. The authors consider groups \Fam{G} of matrices of the form $\left[\small{\begin{smallmatrix} G & 0\\ 0& z \end{smallmatrix}}\right]$, where $z$ is a complex number, and the matrices $G$ form an irreducible subgroup of $\GL(\C)$. When \Fam{G} is not essentially finite, the authors prove that for each invertible $A$ the set $\Fam{G}A$ contains a matrix with more than one eigenvalue. The authors also consider groups $\Fam{G}$ of matrices of the form $\left[\small{\begin{smallmatrix} G & x\\ 0& 1 \end{smallmatrix}}\right]$, where the matrices $G$ comprise a bounded irreducible subgroup of $\GL(\C)$. When \Fam{G} is not finite, the authors prove that for each invertible $A$ the set $\Fam{G}A$ contains a matrix with more than one eigenvalue.

We introduce an equivalence relation, which is finer than the left Wiener–Hopf equivalence at infinity for polynomial matrices, and we obtain discrete invariants and a reduced form for this equivalence relation.

In this article, we shall survey some aspects of our recent (or related) constructions of derived equivalences for algebras and rings.

Let Mm,n be the set of all m × n real matrices. A matrix A ∈ Mm,n is said to be row-dense if there are no zeros between two nonzero entries for every row of this matrix. We find the structure of linear functions T: Mm,n → Mm,n that preserve or strongly preserve row-dense matrices, i.e., T(A) is row-dense whenever A is row-dense or T(A) is row-dense if and only if A is row-dense, respectively. Similarly, a matrix A ∈ Mn,m is called a column-dense matrix if every column of A is a column-dense vector. At the end, the structure of linear preservers (strong linear preservers) of column-dense matrices is found.

The direct, gauge and projective equivalence problems for one-dimensional second-order linear finite difference operators are solved using the method of equivariant moving frames.

We show that the weak and strong forms of the Generalized Spectral Conjecture (GSC) of Boyle and Handelman are equivalent. The GSC asserts that well understood necessary spectral conditions on a square matrix A over a subring S of the reals are sufficient for that matrix to be shift equivalent over S (in the weak form) or strong shift equivalent over S (in the strong form) to a primitive matrix over S. The foundation of this work is the recent result that the group NK_1(S) of algebraic K-theory exactly captures the refinement of shift equivalence over S by strong shift equivalence over S. The GSC remains open in general even in the case that S equals the real numbers., Comment: The purpose of this repost is the addition of Appendix A: Correction

Multivariate statistics (pertaining to the joint behavior of multiple variables) requires the use of linear (“matrix”) algebra as a notational and computational tool. This Chapter contains a brief overview of the elements of linear algebra needed for the remainder of Part III of this book, and their use in representing random vectors and matrices.

The chapter analyses the difficulties that arise in the formation of correlation matrices for input and output signals in the latent period of an emergency state of industrial facilities, showing that due to significant errors in the estimates of their elements caused by the noise correlated with the useful signal, the use of traditional methods in most cases does not ensure adequate results of solving of many applied problems. In many real industrial facilities, input and output variables are usually various physical quantities and additional errors arise during the formation of normalized correlation matrices, which also disrupts the adequacy of the obtained results. Technologies for forming correlation matrices equivalent to the correlation matrices of useful signals are proposed. This also eliminates the errors caused by the normalization of elements of equivalent normalized correlation matrices. In addition, the possibility of forming correlation matrices of noisy signals equivalent to the matrices of their useful signals by correcting the samples of the analyzed signals by means of samples of equivalent noises is considered. It is also shown that equivalent matrices, in addition to controlling the beginning and dynamics of changes in the emergency state of facilities, also improve the adequacy of the results of solving identification and management problems in various areas of industry, power engineering, transport, etc.

We note that the deep results of Grunewald and Segal on algorithmic problems for arithmetic groups imply the decidability of several matrix equivalence problems involving poset-blocked matrices over Z. Consequently, results of Eilers, Restorff, Ruiz and S{\o}rensen imply that isomorphism and stable isomorphism of unital graph C*-algebras (including the Cuntz-Krieger algebras) are decidable. One can also decide flow equivalence for shifts of finite type, and isomorphism of Z-quiver representations (i.e., finite diagrams of homomorphisms of finitely generated abelian groups)., Comment: 12 pages. This version (2) to appear in Proceedings AMS

Let A be an n X ra complex matrix. Let Sim (A) denote the similarity equivalence class of A, Conj(.4) denote the conjunctivity equivalence class of .4, UEquiv(.4) denote the unitaryequivalence equivalence class of .4, and 2/{{A) denote the unitary similarity equivalence class of A. Each of these equivalence classes has been studied for some time and is generally wellunderstood. In particular, canonical forms have been given for each equivalence class. Since the intersection of any two equivalence classes of .4 is again an equivalence class of .4, we consider two such intersections: CS(.4) = Sim(.4) fl Conj(.4) and UES(.4) = Sim(A) n UEquiv(.4). Though it is natural to first think that each of these is simply U{A), for each .4. we show by examples that this is not the case. We then try to classify which matrices .4 have CS(.4) = U{A). For matrices having CS(.4) ^ 1({A), we try to count the number of disjoint unitary similarity classes contained in CS(.4). Though the problem is not completely solved for CS(.4). we reduce the problem to non-singular, non-co-Hermitian matrices .4. A similar analysis is performed for UES(.4), and a (less simple) reduction of the problem is also achieved.

In this work we present a neighbourhood-based image comparison algorithm that makes use of the class of matrix resemblance functions. Based on the concepts of restricted equivalence functions and inclusion grade in the terms of Sinha and Dougherty, we discuss two construction methods for matrix equivalence functions. Moreover, we study the conditions under which this class of functions satisfy a set of properties that are favourable for image comparison operators. We conclude this work with an instance of a potential application for this proposal in video motion detection.

An algebra with a basis consisting solely of units has been called an invertible algebra, and such bases are said to be invertible bases. We announce necessary and sufficient conditions for Leavitt path algebras to be invertible, illustrate the use of the announced criteria, and provide examples of invertible bases.

In the study of local discrimination for multipartite unitary operations, Duan et al. (Phys Rev Lett 100(2):020503, 2008) exhibited an ingenious expression: Any two different unitary operations $$U_1$$U1 and $$U_2$$U2 are perfectly distinguishable by local operations and classical communication in the single-run scenario if and only if 0 is in the local numerical range of $$U_1^\dag U_2$$U1?U2. However, how to determine when 0 is in the local numerical range remains unclear. So it is generally hard to decide the local discrimination of nonlocal unitary operations with a single run. In this paper, for two-qubit diagonal unitary matrices V and their local unitary equivalent matrices, we present a necessary and sufficient condition for determining whether the local numerical range is a convex set or not. The result can be used to easily judge the locally perfect distinguishability of any two unitary operations $$U_1$$U1 and $$U_2$$U2 satisfying $$U_1^\dag U_2=V$$U1?U2=V. Moreover, we design the corresponding protocol of local discrimination. Meanwhile, an interesting phenomenon is discovered: Under certain conditions with a single run, $$U_1$$U1 and $$U_2$$U2 such that $$U_1^\dag U_2=V$$U1?U2=V are locally distinguishable with certainty if and only if they are perfectly distinguishable by global operations.

In the paper, approximate ready trace equivalence for differential semi-algebraic hybrid system is proposed. The equivalence can be used to optimize differential semialgebraic hybrid system. The Concept is proposed on the basis of concrete process algebra and numerical analysis theory. In the approximate ready trace equivalence definition, we consider a cut operator for a polynomial and partial approximation for polynomial. Then we get a strict equivalence between two polynomials. Its advantage is that the new polynomial approximation method overcomes the drawback that traditional approximation method is not transitive, which can be used for automatic reasoning. In order to judge the two differential semi-algebraic hybrid system is equivalent, the axiom system for the approximate ready trace equivalence of differential semi-algebraic hybrid system is presented. This axiom system is a complete axiom system.

We consider the equivalence relations on induced by the Banach subspaces . We show that if , then there is no Borel reduction from the equivalence relation , where X is a Banach space, to .

Let be the ring of all matrices over a field . It is proved that a map is additive if and only if for all , the set of all invertible matrices over .

The equivalence of system is an important concept in multidimensional ($$n$$nD) system, which is closely related to equivalence of multivariate polynomial matrices. This paper mainly investigates the equivalence of some $$n$$nD polynomial matrices, several new results and conditions on the reduction by equivalence of a given $$n$$nD polynomial matrix to its Smith form are obtained.

We discuss the equivalence problem between linear 2D discrete repetitive processes and linear 2D discrete Roesser models. Within the constructive algebraic analysis approach to multidimensional (nD) linear systems theory, this equivalence issue is studied by means of isomorphisms of finitely presented modules. In this paper, we prove that every linear 2D discrete repetitive process is equivalent to an explicit linear 2D discrete Roesser model. Comparing this result to [4], [5] where input / output equivalence is concerned, we point out the differences between these two algebraic notions of equivalence while each notion has interesting applications in nD systems theory.

Two words u and v are said to be k-abelian equivalent if, for each word x of length at most k, the number of occurrences of x as a factor of u is the same as for v. We study some combinatorial properties of k-abelian equivalence classes. Our starting point is a characterization of k-abelian equivalence by rewriting, so-called k-switching. We show that the set of lexicographically least representatives of equivalence classes is a regular language. From this we infer that the sequence of the numbers of equivalence classes is $$\mathbb {N}$$-rational. We also show that the set of words defining k-abelian singleton classes is regular.

It is known that if two matrices are of same size, there may be the equivalent, similar or congruent relations between matrices. This paper has mainly launches the research about relations between matrices of the same size, different size by matrix operations. Such as there is an association between matrix and its adjoint matrix, adjoint matrix is linear combination of power of A. Using a mathematical formula to unify the three equivalence relations.

We describe the inner models with representatives in all equivalence classes of thin equivalence relations in a given projective pointclass of even level assuming projective determinacy. The main result shows that these models are characterized by their correctness and the property that they correctly compute the tree from the appropriate scale. The main step towards this characterization shows that the tree from a scale can be reconstructed in a generic extension of an iterate of a mouse. We then construct models with this property as generic extensions of iterates of mice under the assumption that the corresponding projective ordinal is below ω 2 . On the way, we consider several related problems, including the question when forcing does not add equivalence classes to thin projective equivalence relations. For instance, we show that if every set has a sharp, then reasonable forcing does not add equivalence classes to thin provably Δ 3 1 equivalence relations, and generalize this to all projective levels.

We begin with a paradox of the equivalence relation, and we solve it by using the neutral value of the negation. The so-called Pliant equivalence operator fulfils the modified requirements of the fuzzy equivalence relations. After, we study two models of the equivalence operator. We show that in the Pliant operator case the natural extension of two expressions is equivalent. It has two different types of transitivity. It is associative, and it can be extended to many variables. On this basis, we can create the weighted form of the equivalence operator.

In order to further integrate the content of Linear Algebra, and deeply reveal the equivalence relationship of matrix, this paper discusses the three equivalence relationships on the set of matrices: matrix equivalence、matrix similarity and matrix contract, and gives the corresponding equivalence classes, which further enriches the theory of Linear Algebra.

In this paper, we consider an operator for refinement of variables to b e used in the design of hybrid system. Variables on a given level of abstraction are replaced by more complicated processes on a lower level just like the function are called in the program. Then we established the equivalence, bisimulation equivalence and approximate bisimulation equivalence which are by polynomial flow event structures. These equivalence, bisimulation equivalence a nd approximate bisimulation equivalence are based on the common forms of their zeros. The example show that the equivalence, bisimulation equivalence and approximate bisimulation equivalence are preserved or not under the variables refinement, if the equivalence is not preserved precisely, then we can use the approximate methods to make them approximate equivalence. Lastly we show that our refineme nt has some nice properties.

The article discusses the set of square n × n binary matrices with the same number of 1’s in each row and each column. An equivalence relation on this set is introduced. Each binary matrix is represented using ordered n-tuples of natural numbers. We are looking for a formula which calculates the number of elements of each factor-set by the introduced equivalence relation. We show a relationship between some particular values of the parameters and the Fibonacci sequence.

We consider left-invariant control affine systems, evolving on three-dimensional matrix Lie groups. Equivalence and controllability are investigated. All full-rank systems are classified, under detached feedback equivalence. A representative is identified for each equivalence class. The controllability nature of these representatives is determined.

Let be the algebra of all complex matrices. We characterize linear maps satisfyingfor all , .

We concern with the classification problem of generalized linear control systems, which might be useful for some engineering applications. Inspired by the work of Shayman and Zhou in 1987, we give the definition of linear, differentiable and topological equivalence for a special class of generalized linear systems in a unified way. Then we derive some properties on these three kinds of equivalences and show the canonical forms through a concrete example. In this paper, we obtain natural generalizationsof the Brunovsky's feedback equivalence theorem and the Willems' topological classification theorem for usual linear control systems.

For a matrix pencil A 0 x − A 1 , where A 0 and A 1 are (n × n) -matrices over an arbitrary field F , and A0 is a nonsingular matrix, we establish the normal form with respect to semiscalar equivalence. We also describe the structure of nonsingular polynomial matrices over the field F , which can be reduced to the established form by the transformations of semiscalar equivalence.

In this paper using the second order formula, namely the ∀∃(∀)-identities, we characterize some subclasses of the invertible algebras that are linear over an Abelian group and have restrictions on the use of the automorphisms of the corresponding group.

In this article, the invertible matrices over commutative semirings are studied. Some properties and equivalent descriptions of the invertible matrices are given and the inverse matrix of an invertible matrix is presented by analogues of the classic adjoint matrix. Also, Cramer's rule over a commutative semiring is established. The main results obtained in this article generalize the corresponding results for matrices over commutative rings, for lattice matrices, for incline matrices, for matrices over zerosumfree semirings and for matrices over additively regular semirings.

We study the notion of polynomial-time relation reducibility among computable equivalence relations. We identify some benchmark equivalence relations and show that the reducibility hierarchy has a rich structure. Specifically, we embed the partial order of all polynomial-time computable sets into the polynomial-time relation reducibility hierarchy between two benchmark equivalence relations $\mathsf{E}_{\lambda}$ and $\mathsf{id}$ . In addition, we consider equivalence relations with finitely many nontrivial equivalence classes and those whose equivalence classes are all finite.