Showing papers on "Square matrix published in 2019"
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19 Sep 2019
TL;DR: Inverse solvers of linear equations as discussed by the authors have been used to solve the problem of solving linear equations in the special case of "square" systems, such as the one described in this paper.
Abstract: THE IDEA OF INVERSE Solving Systems of Linear Equations The Special Case of "Square" Systems GENERATING INVERTIBLE MATRICES A Brief Review of Gauss Elimination with Back Substitution Elementary Matrices The LU and LDU Factorization The Adjugate of a Matrix The Frame Algorithm and the Cayley-Hamilton Theorem SUBSPACES ASSOCIATED TO MATRICES Fundamental Subspaces A Deeper Look at Rank Direct Sums and Idempotents The Index of a Square Matrix Left and Right Inverses THE MOORE-PENROSE INVERSE Row Reduced Echelon Form and Matrix Equivalence The Hermite Echelon Form Full Rank Factorization The Moore-Penrose Inverse Solving Systems of Linear Equations Schur Complements Again GENERALIZED INVERSES The {1}-Inverse {1,2}-Inverses Constructing Other Generalized Inverses {2}-Inverses The Drazin Inverse The Group Inverse NORMS The Normed Linear Space Cn Matrix Norms INNER PRODUCTS The Inner Product Space Cn Orthogonal Sets of Vectors in Cn QR Factorization A Fundamental Theorem of Linear Algebra Minimum Norm Solutions Least Squares PROJECTIONS Orthogonal Projections The Geometry of Subspaces and the Algebra of Projections The Fundamental Projections of a Matrix Full Rank Factorizations of Projections Affine Projections Quotient Spaces SPECTRAL THEORY Eigenstuff The Spectral Theorem The Square Root and Polar Decomposition Theorems MATRIX DIAGONALIZATION Diagonalization with Respect to Equivalence Diagonalization with Respect to Similarity Diagonalization with Respect to a Unitary The Singular Value Decomposition JORDAN CANONICAL FORM Jordan Form and Generalized Eigenvectors The Smith Normal Form MULTILINEAR MATTERS Bilinear Forms Matrices Associated to Bilinear Forms Orthogonality Symmetric Bilinear Forms Congruence and Symmetric Matrices Skew-Symmetric Bilinear Forms Tensor Products of Matrices APPENDIX A: COMPLEX NUMBERS What is a Scalar? The System of Complex Numbers The Rules of Arithmetic in C Complex Conjugation, Modulus, and Distance The Polar Form of Complex Numbers Polynomials over C Postscript APPENDIX B: BASIC MATRIX OPERATIONS Introduction Matrix Addition Scalar Multiplication Matrix Multiplication Transpose Submatrices APPENDIX C: DETERMINANTS Motivation Defining Determinants Some Theorems about Determinants The Trace of a Square Matrix APPENDIX D: A REVIEW OF BASICS Spanning Linear Independence Basis and Dimension Change of Basis INDEX
51 citations
TL;DR: In this paper, the authors partially integrate a system of rectangular matrix Riccati equations which describe the synchronization behavior of a nonlinear complex system of N globally connected oscillators, and show by numerical example that complete synchronization can occur even for the mixed case.
Abstract: We partially integrate a system of rectangular matrix Riccati equations which describe the synchronization behavior of a nonlinear complex system of N globally connected oscillators. The equations take a restricted form in which the time-dependent matrix coefficients are independent of the node. We use linear fractional transformations to perform the partial integration, resulting in a system of reduced size which is independent of N, generalizing the well-known Watanabe-Strogatz reduction for the Kuramoto model. For square matrices, the resulting constants of motion are related to the eigenvalues of matrix cross ratios, which we show satisfy various properties such as symmetry relations. For square matrices, the variables can be regarded as elements of a classical Lie group, not necessarily compact, satisfying the matrix Riccati equations. Trajectories lie either within or on the boundary of a classical domain, and we show by numerical example that complete synchronization can occur even for the mixed case. Provided that certain unitarity conditions are satisfied, we extend the definition of cross ratios to rectangular matrix systems and show that again the eigenvalues are conserved. Special cases are models with real vector unknowns for which trajectories lie on the unit sphere in higher dimensions, with well-known synchronization behavior, and models with complex vector wavefunctions that describe synchronization in quantum systems, possibly infinite-dimensional.We partially integrate a system of rectangular matrix Riccati equations which describe the synchronization behavior of a nonlinear complex system of N globally connected oscillators. The equations take a restricted form in which the time-dependent matrix coefficients are independent of the node. We use linear fractional transformations to perform the partial integration, resulting in a system of reduced size which is independent of N, generalizing the well-known Watanabe-Strogatz reduction for the Kuramoto model. For square matrices, the resulting constants of motion are related to the eigenvalues of matrix cross ratios, which we show satisfy various properties such as symmetry relations. For square matrices, the variables can be regarded as elements of a classical Lie group, not necessarily compact, satisfying the matrix Riccati equations. Trajectories lie either within or on the boundary of a classical domain, and we show by numerical example that complete synchronization can occur even for the mixed ca...
43 citations
TL;DR: The condition numbers of each of the three methods are robust with respect to all parameters involved, including the mesh parameter, therefore, the preconditioners are suitable for a variety of problems where such matrix structures arise.
Abstract: Two-by-two block matrices with square matrix blocks arise in many important applications. Since the problems are of large scale, iterative solution methods must be used. Thereby the choice of an efficient and robust preconditioner is crucial. This paper presents two earlier used such preconditioners followed by a novel preconditioner based on transforming the given matrix to a proper form. Sharp eigenvalue estimates are derived. The condition numbers of each of the three methods are robust with respect to all parameters involved, including the mesh parameter. Therefore, the preconditioners are suitable for a variety of problems where such matrix structures arise. The performance of the methods are also compared numerically on a set of test problems.
25 citations
TL;DR: In this paper, a semi-tensor product (STP) of matrices is introduced, which can be seen as an extension of the classical matrix product to two arbitrary matrices.
Abstract: 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.
21 citations
TL;DR: The proposed NNT-ZNN model with two properties of nonlinear and noise-tolerant for the time-varying and static matrix square root finding fully takes error caused by possible noise on ZNN hardware implementation into account.
Abstract: Based on the indefinite error-monitoring function, we propose a novel Zhang neural network (ZNN) model called NNT-ZNN with two properties of nonlinear and noise-tolerant for the time-varying and static matrix square root finding in this paper. Compared to the existing models associated with the square matrix root finding, the NNT-ZNN model proposed in this study fully takes error caused by possible noise on ZNN hardware implementation into account. Under the background that the large model-implementation error, the model still has the ability to converge to the theoretical square root of the given matrix with simulative results illustrated in the paper. For the purpose of comparison, the ZNN model proposed by Zhang et al. is also introduced. Beyond that, the corresponding convergence results of the NNT-ZNN model corresponding to various activation functions, are also shown via time-varying and static positive definite matrix. In the end, the experiments are simulated with MATLAB, which further verifies the availability, effectiveness of the proposed NNT-ZNN model, and robustness against unknown noise.
20 citations
TL;DR: In this paper, it was shown that ∑i≠jTijpppp... ∑ √ √ n,j=1,2, n,n,n is accretive-dissipative.
Abstract: For i,j=1,2,…,n, let Tij be square matrices of the same size such that the block matrix T=T11T12...T1nT21T22...T2n Tn1Tn2...Tnn is accretive-dissipative. It is shown that ∑i≠jTijpp...
17 citations
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TL;DR: First-order perturbation analysis of a simple eigenvalue and the corresponding right and left eigenvectors of a general square matrix, not assumed to be Hermitian or normal, is presented.
Abstract: We present first-order perturbation analysis of a simple eigenvalue and the corresponding right and left eigenvectors of a general square matrix, not assumed to be Hermitian or normal. The eigenvalue result is well known to a broad scientific community. The treatment of eigenvectors is more complicated, with a perturbation theory that is not so well known outside a community of specialists. We give two different proofs of the main eigenvector perturbation theorem. The first, a block-diagonalization technique inspired by the numerical linear algebra research community and based on the implicit function theorem, has apparently not appeared in the literature in this form. The second, based on complex function theory and on eigenprojectors, as is standard in analytic perturbation theory, is a simplified version of well-known results in the literature. The second derivation uses a convenient normalization of the right and left eigenvectors defined in terms of the associated eigenprojector, but although this dates back to the 1950s, it is rarely discussed in the literature. We then show how the eigenvector perturbation theory is easily extended to handle other normalizations that are often used in practice. We also explain how to verify the perturbation results computationally. We conclude with some remarks about difficulties introduced by multiple eigenvalues and give references to work on perturbation of invariant subspaces corresponding to multiple or clustered eigenvalues. Throughout the paper we give extensive bibliographic commentary and references for further reading.
17 citations
TL;DR: This article focuses on the case of constant-current loads and studies the theoretical framework of a second order optimization method for analytic loss minimization by taking into account the symmetry properties of Y b u s.
Abstract: In power engineering, the Y b u s is a symmetric N × N square matrix describing a power system network with N buses. By partitioning, manipulating and using its symmetry properties, it is possible to derive the K G L and Y G G M matrices, which are useful to define a loss minimisation dispatch for generators. This article focuses on the case of constant-current loads and studies the theoretical framework of a second order optimization method for analytic loss minimization by taking into account the symmetry properties of Y b u s . We define an appropriate matrix functional of several variables with complex elements and aim to obtain the minimum values of generator voltages.
16 citations
TL;DR: This paper describes the convergence of multiple subdivision schemes in terms of the joint spectral radius of certain square matrices derived from subdivision weights and shows how to avoid this obstacle.
14 citations
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TL;DR: In this article, the authors consider tubes in the Euclidean 3-space whose Gauss map n is of coordinate finite I-type, i.e., the position vector n satisfies the relation δ √ Δ √ In = δ n, where √ I is the Laplace operator with respect to the first fundamental form I of the surface and δ is a square matrix of order 3.
Abstract: In this paper, we consider tubes in the Euclidean 3-space whose Gauss map n is of coordinate finite I-type, i.e., the position vector n satisfies the relation {\Delta}In = {\Lambda}n, where {\Delta}I is the Laplace operator with respect to the first fundamental form I of the surface and {\Lambda} is a square matrix of order 3. We show that circular cylinders are the only class of surfaces mentioned above of coordinate finite I-type Gauss map.
13 citations
TL;DR: In this article, the authors characterize a class of concordance measures arising from correlations of transformed random variables, which includes Spearman's rho, Blomqvist's beta and van der Waerden's coefficient as special cases.
Abstract: Measures of concordance have been widely used in insurance and risk management to summarize nonlinear dependence among risks modeled by random variables, which Pearson’s correlation coefficient cannot capture. However, popular measures of concordance, such as Spearman’s rho and Blomqvist’s beta, appear as classical correlations of transformed random variables. We characterize a whole class of such concordance measures arising from correlations of transformed random variables, which includes Spearman’s rho, Blomqvist’s beta and van der Waerden’s coefficient as special cases. Compatibility and attainability of square matrices with entries given by such measures are studied—that is, whether a given square matrix of such measures of concordance can be realized for some random vector and how such a random vector can be constructed. Compatibility and attainability of block matrices and hierarchical matrices are also studied due to their practical importance in insurance and risk management. In particular, a subclass of attainable block Spearman’s rho matrices is proposed to compensate for the drawback that Spearman’s rho matrices are in general not attainable for dimensions larger than three. Another result concerns a novel analytical form of the Cholesky factor of block matrices which allows one, for example, to construct random vectors with given block matrices of van der Waerden’s coefficient.
TL;DR: In this paper, a family of iterations for computing the principal square root of a square matrix $A$ using Zolotarev's rational minimax approximants of the square root function is presented.
Abstract: We construct a family of iterations for computing the principal square root of a square matrix $A$ using Zolotarev's rational minimax approximants of the square root function. We show that these ra...
TL;DR: The inverse scattering transform (IST) with nonzero boundary conditions at infinity is developed for a class of 2 × 2 matrix nonlinear Schrodinger-type systems whose reductions include two equation as mentioned in this paper.
Abstract: The inverse scattering transform (IST) with nonzero boundary conditions at infinity is developed for a class of 2 × 2 matrix nonlinear Schrodinger-type systems whose reductions include two equation...
TL;DR: In this article, the authors give a new characterization of algebraically positive matrices, and characterize all tree sign pattern matrices that allow algebraic positivity, and all star and path sign patterns that require algebraically positivity.
01 Jan 2019
TL;DR: The main result of as mentioned in this paper is that the semialgebraic orbit problem is decidable for dimension d ≥ 3, and the main result represents a natural limit to what can be decided (with respect to reachability) about the orbit of a single matrix.
Abstract: The Semialgebraic Orbit Problem is a fundamental reachability question that arises in the analysis of discrete-time linear dynamical systems such as automata, Markov chains, recurrence sequences, and linear while loops. An instance of the problem comprises a dimension d in N, a square matrix A in Q^{d x d}, and semialgebraic source and target sets S,T subseteq R^d. The question is whether there exists x in S and n in N such that A^nx in T.
The main result of this paper is that the Semialgebraic Orbit Problem is decidable for dimension d <= 3. Our decision procedure relies on separation bounds for algebraic numbers as well as a classical result of transcendental number theory - Baker's theorem on linear forms in logarithms of algebraic numbers. We moreover argue that our main result represents a natural limit to what can be decided (with respect to reachability) about the orbit of a single matrix. On the one hand, semialgebraic sets are arguably the largest general class of subsets of R^d for which membership is decidable. On the other hand, previous work has shown that in dimension d=4, giving a decision procedure for the special case of the Orbit Problem with singleton source set S and polytope target set T would entail major breakthroughs in Diophantine approximation.
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TL;DR: In this paper, the authors derived a graph-theoretic approach based on Kirchhoff's Matrix-Tree Theorem for the stability analysis of complex biological systems in a heterogeneous environment, and applied it to well-known ecological models of single species, predator-prey and competition.
Abstract: Threshold values in population dynamics can be formulated as spectral bounds of matrices, determining the dichotomy of population persistence and extinction. For a square matrix $\mu A + Q$, where $A$ is a quasi-positive matrix describing population dispersal among patches in a heterogeneous environment and $Q$ is a diagonal matrix encoding within-patch population dynamics, the monotonicy of its spectral bound with respect to dispersal speed/coupling strength/travel frequency $\mu$ is established via two methods. The first method is an analytic derivation utilizing a graph-theoretic approach based on Kirchhoff's Matrix-Tree Theorem; the second method employs Collatz-Wielandt formula from matrix theory and complex analysis arguments. It turns out that our established result is a slightly strengthen version of Karlin-Altenberg's Theorem, which has previously been discovered independently while investigating reduction principle in evolution biology and evolution dispersal in patchy landscapes. Nevertheless, our result provides a new and effective approach in stability analysis of complex biological systems in a heterogeneous environment. We illustrate this by applying our result to well-known ecological models of single species, predator-prey and competition, and an epidemiological model of susceptible-infected-susceptible (SIS) type. We successfully solve some open problems in the literature of population dynamics.
TL;DR: The eigenproblem for matrices in max-plus algebra describes the steady state of the system, and therefore it has been intensively studied by many authors and an algorithm to compute the eigenvalue and the corresponding eigenvectors of a square matrix in an iterative way is proposed.
Abstract: The eigenproblem for matrices in max-plus algebra describes the steady state of the system, and therefore it has been intensively studied by many authors. In this paper, we propose an algorithm to compute the eigenvalue and the corresponding eigenvectors of a square matrix in an iterative way. The algorithm is extended to compute the nontrivial eigenvectors for Latin squares in max-plus algebra.
TL;DR: It is shown that given access to a subroutine that decides if two weighted undirected graphs are isomorphic, one may deterministically decide the permutation code equivalence provided that the underlying vector spaces intersect trivially with their orthogonal complement with respect to an arbitrary inner product.
Abstract: The paper deals with the problem of deciding if two finite-dimensional linear subspaces over an arbitrary field are identical up to a permutation of the coordinates. This problem is referred to as the permutation code equivalence. We show that given access to a subroutine that decides if two weighted undirected graphs are isomorphic, one may deterministically decide the permutation code equivalence provided that the underlying vector spaces intersect trivially with their orthogonal complement with respect to an arbitrary inner product. Such a class of vector spaces is usually called linear codes with trivial hulls. The reduction is efficient because it essentially boils down to computing the inverse of a square matrix of order the length of the involved codes. Experimental results obtained with randomly drawn binary codes having trivial hulls show that permutation code equivalence can be decided in a few minutes for lengths up to 50,000.
TL;DR: This paper proposes a novel approach of analog circuit soft fault diagnosis utilizing matrix perturbation analysis, which establishes an output response square matrix whose elements will change if circuits fail.
Abstract: This paper proposes a novel approach of analog circuit soft fault diagnosis utilizing matrix perturbation analysis. This method establishes an output response square matrix whose elements will change if circuits fail. Fault can be diagnosed via comparing the difference between the fault-free output matrix and the faulty. According to matrix theory, matrix spectral radius and perturbation matrix m1 norm are utilized to describe the difference. Differing from artificial intelligence algorithms, it is all completely unnecessary to train samples, and can be applied to more complex circuit diagnostics with fewer test nodes. Fault diagnosis, fault location and parameter identification can be realized by quadratic curve fitting in single fault mode. Experiments confirm the feasibility and correctness of this method.
TL;DR: In this paper, a necessary and sufficient condition in terms of a linear matrix inequality (LMI), where the coefficients A, B appear as a linear function, was introduced to deal with the robust strong stability problem with norm bounded uncertainty, whose solutions are expressed by LMIs.
TL;DR: In this paper, the authors developed a 3D nonlinear Stokes-Mueller polarimetric formalism for the acquisition of a complete complex valued 3D susceptibility tensor of a material.
Abstract: The formalism is developed for a three-dimensional (3D) nonlinear Stokes–Mueller polarimetry that describes a method of acquiring a complete complex valued 3D nonlinear susceptibility tensor of a material. The expressions are derived for generalized 3D linear and nonlinear Stokes vectors and the corresponding nonlinear Mueller matrix. The coherency-like Hermitian square matrix X of susceptibilities is introduced, which is derived from the nonlinear Mueller matrix. The X-matrix is characterized by the index of depolarization. Several decompositions of the X-matrix are introduced that provide a possibility to obtain nonlinear susceptibility tensors of constituting materials in the heterogeneous media. The 3D nonlinear Stokes–Mueller polarimetry formalism can be applied for three and higher wave mixing processes. The 3D polarimetric measurements can be used for structural investigations of materials, including heterogeneous biological structures. The 3D polarimetry is applicable for nonlinear microscopy with high numerical aperture objectives.
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TL;DR: In this paper, the authors give various characterizations of random walks with possibly different steps that have relatively large discrepancy from the uniform distribution modulo a prime p, and use these results to study the distribution of the rank of random matrices over F_p and the equi-distribution behavior of normal vectors of random hyperplanes.
Abstract: In this note we give various characterizations of random walks with possibly different steps that have relatively large discrepancy from the uniform distribution modulo a prime p, and use these results to study the distribution of the rank of random matrices over F_p and the equi-distribution behavior of normal vectors of random hyperplanes. We also study the probability that a random square matrix is eigenvalue-free, or when its characteristic polynomial is divisible by a given irreducible polynomial in the limit n to infinity in F_p. We show that these statistics are universal, extending results of Stong and Neumann-Praeger beyond the uniform model.
TL;DR: In this paper, the convergence results for block Arnoldi and block GMRES have been established for the case of multiple right-hand sides, where the matrices and righthand sides producing any admissible behavior can be constructed.
Abstract: It is well-established that any non-increasing convergence curve is possible for GMRES and a family of pairs $(A,b)$ can be constructed for which GMRES exhibits a given convergence curve with $A$ having arbitrary spectrum. No analog of this result has been established for block GMRES, wherein multiple right-hand sides are considered. By reframing the problem as a single linear system over a ring of square matrices, we develop convergence results for block Arnoldi and block GMRES. In particular, we show what convergence behavior is admissible for block GMRES and how the matrices and right-hand sides producing any admissible behavior can be constructed. Moreover, we show that the convergence of the block Arnoldi method for eigenvalue approximation can be almost fully independent of the convergence of block GMRES for the same coefficient matrix and the same starting vectors.
TL;DR: In this paper, all irreducible 3-by-3 sign pattern matrices are classified into three groups (i) those that require AP, (ii) those which allow AP but not require AP and (iii) those (i.e., not AP) that do not allow AP.
Abstract: A square matrix M with real entries is algebraically positive (AP) if there exists a real polynomial p such that all entries of the matrix p(M) are positive. A square sign pattern matrix S allows algebraic positivity if there is an algebraically positive matrix M whose sign pattern is S. On the other hand, S requires algebraic positivity if matrix M, having sign pattern S, is algebraically positive. Motivated by open problems raised in a work of Kirkland, Qiao, and Zhan (2016) on AP matrices, all nonequivalent irreducible 3 by 3 sign pattern matrices are listed and classify into three groups (i) those that require AP, (ii) those that allow but not require AP, or (iii) those that do not allow AP. A necessary condition for an irreducible n by n sign pattern to allow algebraic positivity is also provided.
TL;DR: In this paper, the authors considered the problem of numerically solving linear algebraic equations with positive definite symmetric matrices or oscillation-type matrices using regularization, which lead to reduced condition numbers.
Abstract: Systems of linear algebraic equations (SLAEs) are considered in this work. If the matrix of a system is nonsingular, a unique solution of the system exists. In the singular case, the system can have no solution or infinitely many solutions. In this case, the notion of a normal solution is introduced. The case of a nonsingular square matrix can be theoretically regarded as good in the sense of solution existence and uniqueness. However, in the theory of computational methods, nonsingular matrices are divided into two categories: ill-conditioned and well-conditioned matrices. A matrix is ill-conditioned if the solution of the system of equations is practically unstable. An important characteristic of the practical solution stability for a system of linear equations is the condition number. Regularization methods are usually applied to obtain a reliable solution. A common strategy is to use Tikhonov’s stabilizer or its modifications or to represent the required solution as the orthogonal sum of two vectors of which one vector is determined in a stable fashion, while seeking the second one requires a stabilization procedure. Methods for numerically solving SLAEs with positive definite symmetric matrices or oscillation-type matrices using regularization are considered in this work, which lead to SLAEs with reduced condition numbers.
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TL;DR: In this article, it was shown that the line integral of the Jacobian of a nonlinear system is an oscillatory matrix, and several new sufficient conditions for such matrices were derived.
Abstract: A matrix $A$ is called totally positive (TP) if all its minors are positive, and totally nonnegative (TN) if all its minors are nonnegative. A square matrix $A$ is called oscillatory if it is TN and some power of $A$ is TP. A linear time-varying system is called an oscillatory discrete-time system (ODTS) if the matrix defining its evolution at each time $k$ is oscillatory. We analyze the properties of $n$-dimensional time-varying nonlinear discrete-time systems whose variational system is an ODTS, and show that they have a well-ordered behavior. More precisely, if the nonlinear system is time-varying and $T$-periodic then any trajectory either leaves any compact set or converges to an $(n-1)T$-periodic trajectory, that is, a subharmonic trajectory. These results hold for any dimension $n$. The analysis of such systems requires establishing that a line integral of the Jacobian of the nonlinear system is an oscillatory matrix. This is non-trivial, as the sum of two oscillatory matrices is not necessarily oscillatory, and this carries over to integrals. We derive several new sufficient conditions guaranteeing that the line integral of a matrix is oscillatory, and demonstrate how this yields interesting classes of discrete-time nonlinear systems that admit a well-ordered behavior.
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TL;DR: In this paper, a Sinkhorn-type algorithm for producing quantum permutation matrices encoding symmetries of graphs is introduced. But it is not yet known whether a given finite graph (without multiple edges) has Banica.
Abstract: We introduce a Sinkhorn-type algorithm for producing quantum permutation matrices encoding symmetries of graphs. Our algorithm generates square matrices whose entries are orthogonal projections onto one-dimensional subspaces satisfying a set of linear relations. We use it for experiments on the representation theory of the quantum permutation group and quantum subgroups of it. We apply it to the question whether a given finite graph (without multiple edges) has quantum symmetries in the sense of Banica. In order to do so, we run our Sinkhorn algorithm and check whether or not the resulting projections commute. We discuss the produced data and some questions for future research arising from it.
TL;DR: It is proved that in every solvable two-sided max-linear system of minimally active or essential type, all positions in C active in any optimal permutation for the assignment problem for C, are also active for some non-trivial solution of the two- sided system.
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TL;DR: A special class of Jordan algebras over a field of characteristic zero is considered in this paper, where the identity matrix, the all-one matrix, Schur-Hadamard multiplication, and the Jordan product are considered.
Abstract: A special class of Jordan algebras over a field $F$ of characteristic zero is considered. Such an algebra consists of an $r$-dimensional subspace of the vector space of all square matrices of a fixed order $n$ over $F$. It contains the identity matrix, the all-one matrix; it is closed with respect to \correction{matrix transposition}, Schur-Hadamard (entrywise) multiplication and the Jordan product $A*B=\frac 12 (AB+BA)$, where $AB$ is the usual matrix product.
The suggested axiomatics (with some natural additional requirements) implies an equivalent reformulation in terms of symmetric binary relations on a vertex set of cardinality $n$. The appearing graph-theoretical structure is called a Jordan scheme of order $n$ and rank $r$. A significant source of Jordan schemes stems from the symmetrization of association schemes. Each such structure is called a non-proper Jordan scheme. The question about the existence of proper Jordan schemes was posed a few times by Peter J. Cameron.
In the current text an affirmative answer to this question is given. The first small examples presented here have orders $n=15,24,40$. Infinite classes of proper Jordan schemes of rank 5 and larger are introduced. A prolific construction for schemes of rank 5 and order $n=\binom{3^d+1}{2}$, $d\in {\mathbb N}$, is outlined.
The text is written in the style of an essay. The long exposition relies on initial computer experiments, a large amount of diagrams, and finally is supported by a number of patterns of general theoretical reasonings. The essay contains also a historical survey and an extensive bibliography.
20 Nov 2019
TL;DR: In this article, the associate matrix of a quaternion square matrix (whose entries are quaternions) is defined as a matrix matrix, and the identity of the associated matrix is obtained by matrix methods.
Abstract: In this paper, we define the associate matrix as% \begin{equation*} F=\left( \begin{array}{cc} 1+i+2j+3k & i+j+2k \\ i+j+2k & 1+j+k% \end{array}% \right) . \end{equation*}% By the means of the matrix $F,$ we give several identities about Fibonacci and Lucas quaternions by matrix methods. Since there are two different determinant definitions of a quaternion square matrix (whose entries are quaternions), we obtain different Cassini identities for Fibonacci and Lucas quaternions apart from Cassini identities that given in the papers \cite% {halici} and \cite{akyigit2}.