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The Theory of Matrices
... Notons qu'une matrice inversible A admet toujours une racine p-ième (ou plus). Cependant, dans le cas d'une matrice non inversible, l'existence des racines p-ièmes dépend de la structure des diviseursélémentaires de cette matrice, correspondantà sa valeur propre nulle (voir [51] et [118]). ...
... L'extension de la notion d'une fonction scalaire f (z)à celle de la fonction de matrices propriété, des fonctions de matrices, sera utilisée dans les chapitres suivants. Les outils présentés dans cette section, peuventêtre trouvés dans les références [51], [56], [62], [101] et [117]. ...
... , s, où m k est la multiplicité de λ k dans le polynôme minimal (m k est en outre la taille du plus grand bloc de Jordan associéà λ k ). Une fois cette condition est vérifiée par f , il est possible de donner un sensà f (A), i.e., définir une fonction de matrices A → f (A) [51]. ...
... S nrk = diag e jθ 1 , e jθ 2 , e jθ 3 , e jθ 4 · S nr · diag e jθ 1 , e jθ 2 , e jθ 3 , e jθ 4 (17) I is the 4 × 4 Identity Matrix, and diag is a 4 × 4 Diagonal Matrix with entries contained in the brackets. Equations (15) represent a system of four equations in four unknowns, θ 1 , θ 2 , θ 3 , θ 4 , which can be solved numerically. ...
... Equations (15) represent a system of four equations in four unknowns, θ 1 , θ 2 , θ 3 , θ 4 , which can be solved numerically. Equations (15)- (17) replace the condition on the S-matrix (or Z-matrix) discussed in [12] for the reciprocal case. ...
... Once the values of θ 1 , θ 2 , θ 3 , θ 4 have been obtained, ζ nrk can be extracted by Equations (16)- (17). From Equation (7), CCVS's values and the reciprocal Z-matrix of the polygon network in Fig. 2 ...
In this paper, a technique to identify/synthesize an equivalent circuit of nonreciprocal lossy N-port device is presented. The technique joins the classical procedure discussed in the '60s to the polygon network recently proposed in the literature, which permits to draw an equivalent circuit for reciprocal lossless N-port device in a very simple way. The identification is applied to two microwave devices, a reciprocal lossy iris in WR90 waveguide and a 3-port nonreciprocal lossy circulator. The proposed equivalent circuit could give some information about the agreement of the manufactured device and its design, which usually is developed in the hypothesis of ideal lossless components.
... Melalui konsep keteramatan, keadaan awal dari suatu sistem dapat diketahui hanya dengan bermodalkan informasi terkini dari sistem. Dalam makalah ini keteramatan sistem deskriptor akan dibahas berdasarkan konsep dan karakterisasi yang disajikan oleh Dai dan Yip [3], [5]. ...
... [5] Sistem deskriptor ( ) mempunyai solusi jika matriks pensil ( , ) regular, yaitu | + | ≠ 0 Selanjutnya, diberikan lemma yang sangat bermanfaat untuk menentukan regularitas suatu sistem deskriptor.Lemma 2.3. [2],[3],[4] pensil ( , ) regular jika dan hanya jika dapat dipilih suatu matriks nonsingular ...
In this paper the observability of continuous descriptor system of the form Ex(t)= Ax(t) Bu(t), x(0)=x0 will be studied, where E,A, and B are constant matrices that may be singular and u(t) is piecewise continuous function which is differentiated (m-1) times, where m is the degree of nilpotency system. Two definitions about observability of descriptor systems along with their characterizations given by Dai and Yip will be both discussed, then further the relationship and comparison between these characterizations will be presented.
... So, for the element g = n i=1 α i g i in the theorem, the matrix G = A g is a non-negative matrix with constant column sum r = n i=1 α i . It follows that G is a multiple of a column stochastic matrix, with eigenvalue of greatest modulus r (see [1]). Now, suppose that Pt = =t − λ 1 · · · ·t − λ k = t k + c 1 t k−1 + · · · + c k is a factor of the characteristic polynomial of G which is irreducible over the rational field Q. ...
... where G 1 is, say, an i × i submatrix while G 2 is a j − i × ×j − i submatrix with 0 < i < j (see [1]). Indeed, the permutations in question can be obtained simply by a relabeling of the elements of J. Thus, we may assume that the elements a 1 a i a i+1 a j of J are listed so that, with respect to the corresponding basis for V J , the matrix G J is in the above block form. ...
It is shown that a set of finite semigroups satisfying certain restrictions on spectra of sums of generators is finite.
... First, we recall a result of Lyapunov [14]: If for any positive definite symmetric matrix ( > 0) there is a negative definite symmetric matrix ( < 0) satisfying In the more general case when the matrix is positive semi-definite ( 0), the triplets In( ) and In( ) do not generally coincide with each other. A useful result for this case involves the concept of controllability of the matrix pair ( , ). ...
... Obviously, the pair ( , ) is controllable (i.e., rank ( | ) = 2 ) if and only if rank( (^| 1 ), (^| 2 )) = and rank(^(^| 1 ), (^| 2 )) = , since the reduction of a matrix by elementary operations does not change its rank. Now, according to the Cayley-Hamilton theorem (see [14]), the matrix^1 can be represented by a linear combination of the matrices 1 ,^1, . . . ,^− 1 1 , and consequently rank( (^| 1 ),^1, (^| 2 )) = rank( (^| 1 ), (^| 2 )). ...
The note is concerned with the problem of determining the completely unstable linear non-conservative undamped (circulatory) dynamical systems. Several conditions that provide the complete instability for such systems are derived using the direct method of Lyapunov and the concept of controllability. The conditions are expressed directly via the matrices describing the dynamical system.
... The transformation P (λ) U (λ)P (λ)V (λ) is called a unimodular equivalence transformation and the canonical form with respect to this transformation is the Smith form[21], recalled in the following theorem. Theorem 2.3.[21]Let ...
... The transformation P (λ) U (λ)P (λ)V (λ) is called a unimodular equivalence transformation and the canonical form with respect to this transformation is the Smith form[21], recalled in the following theorem. Theorem 2.3.[21]Let P (λ) be an m × n matrix polynomial over C. Then there exists r ∈ N, r min{m, n} and unimodular matrix polynomials U (λ) and V (λ) over C such that U (λ)P (λ)V (λ) = g 1 (λ) 0 0 r×(n−r) ...
The set POLd,rm×n of m×n complex matrix polynomials of grade d and (normal) rank at most r in a complex (d+1)mn dimensional space is studied. For r=1,…,min{m,n}−1, we show that POLd,rm×n is the union of the closures of the rd+1 sets of matrix polynomials with rank r, degree exactly d, and explicitly described complete eigenstructures. In addition, for the full-rank rectangular polynomials, i.e. r=min{m,n} and m≠n, we show that POLd,rm×n coincides with the closure of a single set of the polynomials with rank r, degree exactly d, and the described complete eigenstructure. These complete eigenstructures correspond to generic m×n matrix polynomials of grade d and rank at most r.
... which shows that in general M has only two (instead of four) independent eigen- vectors belonging to the eigenvalue 0 and only one eigenvector each (instead of two) corresponding to the eigenvalues ±(η − 3) 1 2 . This means that the Jordan canonical form [50] of M is ...
Summary It is shown that there exists for the quantum harmonic oscillator a large class of « semi-coherent » states, which represent
harmonically oscillating wave packets whose « spread » or « size » remains constant. These states are different from the coherent
states in not having the minimum value 1/2ħ for the uncertainty product Δx·Δp.
... We point out that a similar and elegant proof of the above sufficient condition is provided by Hestenes (1975). There are various proofs of the Sylvester criterion which are based on the Lagrange-Jacobi reduction formula; some proofs are clear and efficient from a didactic point of view, but rather long, such as for example the proofs of Gantmacher (1959), Hadley (1961) and Hohn (1973). Some other proofs are more concise but also more difficult to grasp and therefore not too suitable for a course to undergraduates; it is the case, e. g., of the classical paper of Debreu (1952), of the books of Aleskerov and others (2011) and Murata (1977). ...
In the first part of the paper we present several proofs of the so-called Sylvester criterion for quadratic forms; some of the said proofs are short and easy. In the second part of the paper we give an algebraic proof of the Sylvester criterion for quadratic forms subject to a linear homogeneous system.
... If A is irreducible then y > 0 and is unique up to multiplication by a positive scalar. There are many classical and recent books giving a full account of the PerronFrobenius theory of nonnegative matrices for example [3,12,17,23,29,30,35]. It is well known that PF-theory found innumerous applications in all sciences. ...
In this paper we consider the Collatz-Wielandt quotient for a pair of nonnegative operators A,B that map a given pointed generating cone in the first space to subsets of a given pointed generating cone in the second space. In the case the two spaces and the two cones are identical, and B is the identity operator this quotient is the spectral radius of A. In some applications, as cellular communication and quantum information theory, one needs to deal with the Collatz-Wielandt quotient for two nonnegative operators. In this paper we treat the two important cases: a pair of rectangular nonnegative matrices and a pair completely positive operators. We give a characterization of minimal optimal solutions and polynomially computable bounds on the Collatz-Wielandt quotient.
... which shows that in general M has only two (instead of four) independent eigenvectors belonging to the eigenvalue 0 and only one eigenvector each (instead of two) corresponding to the eigenvalues ±(η − 3) 1 2. This means that the Jordan canonical form [50] of M is ...
... W is primitive (i.e. irreducible and aperiodic) (DeGroot, 1974;Gantmacher, 2000). ...
Investigation of social influence dynamics requires mathematical models that are “simple” enough to admit rigorous analysis, and yet sufficiently “rich” to capture salient features of social groups. Thus, the mechanism of iterative opinion pooling from (DeGroot, 1974), which can explain the generation of consensus, was elaborated in (Friedkin and Johnsen, 1999) to take into account individuals’ ongoing attachments to their initial opinions, or prejudices. The “anchorage” of individuals to their prejudices may disable reaching consensus and cause disagreement in a social influence network. Further elaboration of this model may be achieved by relaxing its restrictive assumption of a time-invariant influence network. During opinion dynamics on an issue, arcs of interpersonal influence may be added or subtracted from the network, and the influence weights assigned by an individual to his/her neighbors may alter. In this paper, we establish new important properties of the (Friedkin and Johnsen, 1999) opinion formation model, and also examine its extension to time-varying social influence networks.
... Using Lyapunov's transform [3] ( , ) ...
Стаття присвячена аналізу усталеного процесу в ланцюзі інвертора. Оскільки конфігурація ланцюга інвертора змінюється, процеси в такому колі описуються диференційними рівняннями зі змінними коефіцієнтами. Передбачається, що ланцюг інвертора змінюється періодично. Для знаходження усталеного процесу використовуються метод розширення звичайного диференційного рівняння до рівняння в частинних похідних, перетворення Ляпунова та двовимірне перетворення Лапласа. Для процесів, що протікають в ланцюзі інвертора, вводиться поняття передавальної функції і частотних характеристик. Встановлений процес отриманий у вигляді подвійного ряду Фур'є. Як приклад розглянуто інвертор, для якого представлені частотні характеристики.Бібл. 6, рис. 6.
... The index ν − (S n ) for a Hankel matrix S n can be calculated by the Frobenius rule (see [27,Theorem X.24]). In particular, if all the determinants D n := det S n (n ∈ Z + ) do not vanish, then ν − (S n ) coincides with the number of sign alternations in the sequence ...
A function f meromorphic on ℂ\ℝ is said to be in the generalized Nevanlinna class Nκ (κ ϵ ℤ+), if f is symmetric with respect to ℝ and the kernel Nωz≔fz−fω¯z−ω¯ has κ negative squares on ℂ+. The generalized Stieltjes class Nκkκk∈ℤ+ is defined as the set of functions f ϵ Nκ such that z f ϵ Nk. The full indefinite Stieltjes moment problem MPκks consists in the following: Given κ, k ϵ ℤ+, and a sequence s=sii=0∞ of real numbers, to describe the set of functions f∈Nκk, which satisfy the asymptotic expansion
fz=−s0z−⋯−s2nz2n+1+o1z2n+1z=−y∈ℝ−y↑∞
for all n big enough. In the present paper, we will solve the indefinite Stieltjes moment problem MPκks within the M. G. Krein theory of u-resolvent matrices applied to a Pontryagin space symmetric operator A[0;N] generated by J0N. The u-resolvent matrices of the operator A[0;N] are calculated in terms of generalized Stieltjes polynomials, by using the boundary triple’s technique. Some criteria for the problem MPκks to be solvable and indeterminate are found. Explicit formulae for Padé approximants for the generalized Stieltjes fraction in terms of generalized Stieltjes polynomials are also presented.
... Lemma 4 (Weierstrass canonical form). A pencil λE − A with A, E ∈ R m×m can be decomposed into a regular and a singular part, denoted by A R and E S , respectively, by the Weierstrass canonical form (Gantmacher, 1960;Gerdin, 2004;Kailath, 1980;Luenberger, 1978) ...
Multidimensional systems are becoming increasingly important as they provide a promising tool for estimation, simulation and control, while going beyond the traditional setting of one-dimensional systems. The analysis of multidimensional systems is linked to multivariate polynomials, and is therefore more difficult than the well-known analysis of one-dimensional systems, which is linked to univariate polynomials. In the current paper we relate the realization theory for overdetermined autonomous multidimensional systems to the problem of solving a system of polynomial equations. We show that basic notions of linear algebra suffice to analyze and solve the problem. The difference equations are associated with a Macaulay matrix formulation, and it is shown that the null space of the Macaulay matrix is a multidimensional observability matrix. Application of the classical shift trick from realization theory allows for the computation of the corresponding system matrices in a multidimensional state-space setting. This reduces the task of solving a system of polynomial equations to computing an eigenvalue decomposition. We study the occurrence of multiple solutions, as well as the existence and analysis of solutions at infinity, which allow for an interpretation in terms of multidimensional descriptor systems.
... where the first equality holds because ∇• is metric, and the third one comes from the condition ∇• ω = 0. We conclude that A∇ X Y = ∇ X AY , and from the fact that J 0 can be written as polynomial in A [17], we obtain ∇• J 0 = 0. Now, using that ω and J 0 are parallel with respect to ∇• we have ...
We consider compatibility conditions between Poisson and Riemannian structures on smooth manifolds by means of a contravariant partially complex structure, or f-structure, introducing the notion of (almost) K\"ahler--Poisson manifolds. In addition, we study some of their properties under structure preserving maps and symmetries.
... for each n ∈ N (see for instance[5]). The following auxiliary result also will be used in the proof of Theorem 1. ...
The relation between the Darboux transformation and the solutions of the full Kostant Toda lattice is analyzed. The discrete Korteweg de Vries equation is used to obtain such solutions and the main result of [1] is extended to the case of general (p+ 2)-banded Hessenberg matrices.
The composition operators preserving total non-negativity and total positivity for various classes of kernels are classified, following three themes. Letting a function act by post composition on kernels with arbitrary domains, it is shown that such a composition operator maps the set of totally non-negative kernels to itself if and only if the function is constant or linear, or just linear if it preserves total positivity. Symmetric kernels are also discussed, with a similar outcome. These classification results are a byproduct of two matrix-completion results and the second theme: an extension of A. M. Whitney’s density theorem from finite domains to subsets of the real line. This extension is derived via a discrete convolution with modulated Gaussian kernels. The third theme consists of analyzing, with tools from harmonic analysis, the preservers of several families of totally non-negative and totally positive kernels with additional structure: continuous Hankel kernels on an interval, Pólya frequency functions, and Pólya frequency sequences. The rigid structure of post-composition transforms of totally positive kernels acting on infinite sets is obtained by combining several specialized situations settled in our present and earlier works.
Requirements are investigated in this paper for each descriptor form subsystem, with which a causal/impulse free networked dynamic system (NDS) can be constructed. For this purpose, a matrix rank-based necessary and sufficient condition is at first derived for the causality/impulse freeness of an NDS, with the matrix depending affinely on subsystem connections. From this result, a necessary and sufficient condition is derived on each subsystem for causal/impulse free NDS constructibility. This condition further leads to a necessary and sufficient condition for the existence of a local static output feedback that guarantees the construction of a causal/impulse free NDS. Remarkably, all the numerical computations involved in these conditions are performed independently on each individual subsystem. Situations have also been clarified in which NDS causality/impulse freeness is independent of subsystem connections. It has also been made clear that local static output feedbacks may not be helpful in constructing a causal NDS.
This work concerns with the existence of traveling wave solutions for the following diffusive predator–prey type system with Holling type-III functional response: where all parameters are positive which will be mentioned later. The traveling wave solutions are established in , which is a heteroclinic orbit connecting the boundary equilibrium and the positive equilibrium. Applying the methods of Wazewski Theorem and LaSalle’s Invariance Principle, and constructing a Liapunov function, we obtain the existence of traveling wave solutions. We also discuss some possible biological implications of the existence of these waves.
Hersh (1997) in a book aptly named What Is Mathematics Really? stresses the great distance he detects between the reality of professional mathematical practice—contemporary and historical—and the reasoning in formal languages that philosophers (since Frege) have largely characterized mathematical proof in terms of. Hersh criticizes the reasoning-in-formal-languages view of mathematical practice and mathematical proof as “isolated,” “timeless,” “ahistorical,” and indeed, even “inhuman.” Hersh (1997, xi) contrasts this derivation-centered view of mathematics (and mathematical proof) with an alternative view that takes mathematics to be a human activity and a social phenomenon, one which historically evolves and is intelligible only in a social context. His alternative view pointedly roots mathematical practice in the actual proofs that mathematicians create—actual proofs that Hersh claims philosophers of mathematics often ignore.
Let X be a matrix with entries in a polynomial ring over an algebraically closed field K. We prove that, if the entries of X outside some -submatrix are algebraically dependent over K, the arithmetical rank of the ideal of t-minors of X drops at least by one with respect to the generic case; under suitable assumptions, it drops at least by k if X has k zero entries. This upper bound turns out to be sharp if , since it then coincides with the lower bound provided by the local cohomological dimension.
R. Guralnick [Linear Algebra Appl. 99, 85-96 (1988)] proved that two holomorphic matrices on a noncompact connected Riemann surface, which are locally holomorphically similar, are globally holomorphically similar. In the preprints [arXiv:1703.09524] and [arXiv:1703.09530], a generalization of this to arbitrary (possibly, nonsmooth) 1-dimensional Stein spaces was obtained. The present paper contains a revised version of the proof from [arXiv:1703.09524]. The method of this revised proof can be used also in the higher dimensional case, which will be the subject of a forthcoming paper.
In this chapter, a strip-method suitable for reducing pulse interference in communication channels, cryptography, steganography, and other applications is considered. The invariants to fragmentation and double-sided matrix transformation of images provide the noise immunity and transmission security. The chapter contains new definitions of invariants, as well as invariant images of the first and second types. Moreover, tasks of analyzing and synthesizing both invariants and corresponding transformation matrices are set forth too. The criteria of their existences are derived and methods for creation of invariant images using eigenvectors of transforming matrices are proposed. Some cases of complex and multiple eigenvalues of a direct transformation matrix are considered. It was proposed to solve the problem of finding the matrices of direct and inverse transformations by means of a given set of invariant images. The solution of the task of arraying the matrix of double-sided transformation according to a given set of invariant images is suggested.
This paper discusses a novel implementable approach to an exact linearization procedure based on the implicit systems techniques. The formal procedure we propose includes a specific "split-ting" of the nonlinear state representation in two parts that involve a basic rectangular representation and an auxiliary nonlinear algebraic equation. The proposed linear implicit systems description makes it possible to apply the conventional linear control techniques to an initially given sophisticated nonlinear dynamic model.
We pose a linear matrix differential-algebraic boundary-value problem generalizing the traditional linear boundary-value problems for differential-algebraic equations. We have found the constructive conditions of existence and an algorithm of construction of the solutions of a linear matrix differentialalgebraic boundary-value problem. We propose the construction of a generalized Green operator for the determination of solutions of the linear differential-algebraic boundary-value problem and give some examples of construction of such solutions.
The structure of a rational matrix is given by its Smith-McMillan invariants. Some properties of the Smith-McMillan invariants of rational matrices with elements in different principal ideal domains are presented: In the ring of polynomials in one indeterminate (global structure), in the local ring at an irreducible polynomial (local structure), and in the ring of proper rational functions (infinite structure). Furthermore, the change of the finite (global and local) and infinite structures is studied when performing a Möbius transformation on a rational matrix. The results are applied to define an equivalence relation in the set of polynomial matrices, with no restriction on size, for which a complete system of invariants are the finite and infinite elementary divisors.
We believe that the difference between time scale systems and ordinary differential equations is not as big as people use to think. We consider linear operators that correspond to linear dynamical systems on time scales. We study solvability of these operators in . For ordinary differential equations such solvability is equivalent to hyperbolicity of the considered linear system. Using this approach and transformations of the time variable, we spread the concept of hyperbolicity to time scale dynamics. We provide some analogs of well-known facts of Hyperbolic Systems Theory, e.g. the Lyapunov -- Perron theorem on stable manifold.
This article presents a concept to define a function of matrix. Start by defining a power function of matrix is then expanded to the concept of the definition of any function of matrix. Some calculating-examples are provided to clearly understand. An application to financial mathematics is shown.
This chapter is dedicated to the development of the optimum control theory, which forms the basis of control systems analysis and design for a large number of problems and those that occur in differential game theory. Theoretical developments are aimed at optimization of a general, non-linear costfunction (performance index) where the evolution of the states is defined by a set of non-linear differential equations. The chapter introduces the use of the Euler-Lagrange multiplier for incorporating equality constraints, and the construction of the Hamiltonian for deriving optimum control strategies for parties involved in a game. It then considers the dynamic optimization problem utilizing the Bolza formulation and uses variational calculus to derive necessary and sufficient conditions for optimality. Finally, the chapter considers the application of the linear system with quadratic performance index (LQPI) problem to two-party and three-party differential game guidance problems. These are recognized as the pursuer-evader games.
General Structure of the Symbolic-Numerical MethodThe Case of Diagonalizable Amplification MatricesScheme CheckerSymbolic Stages of the Method
Generation of a FORTRAN Program by Computer AlgebraComputation of the Coordinates of Points of a Stability Region BoundaryImproved Accuracy of Numerical ResultsExamples of Stability Analyses of Difference Schemes for Equations of Hyperbolic TypeStability Analysis of the MacCormack Scheme for Two-Dimensional Euler EquationsStability Analysis of the MacCormack Scheme for Three-Dimensional Euler EquationsExamples of Stability Analyses of Difference Schemes for Navier-Stokes Equations Stokes Equations
Приводятся необходимые условия перестановочности степеней в конечномерной алгебре над полем.
We develop a wave-function-based method for the simulation of quantum dynamics of systems with many degrees of freedom at finite temperature. The method is inspired by the ideas of Thermo Field Dynamics (TFD). As TFD, our method is based on the doubling of the system's degrees of freedom and thermal Bogoliubov transformation. As distinct from TFD, our method implements the doubling of thermalized degrees of freedom only, and relies upon the explicitly constructed generalized thermal Bogoliubov transformation, which is not restricted to fermionic and bosonic degrees of freedom. This renders the present approach computationally efficient and applicable to a large variety of systems.
The dynamics of the rotational motion of a satellite moving in the central Newtonian force field in a circular orbit under the influence of gravitational and active damping torques is investigated with the help of computer algebra methods. The properties of a nonlinear algebraic system that determines equilibrium orientations of a satellite under the action of gravitational and active damping torques were studied. An algorithm for the construction of a Gröbner basis is proposed for determining the equilibrium orientations of a satellite with given central moments of inertia and given damping torques. The conditions of the equilibria’s existence were obtained by the analysis of real roots of algebraic equations from the constructed Gröbner basis. The domains with an equal number of equilibria were specified by using algebraic methods for the construction of discriminant hypersurfaces. The conditions of asymptotic stability of the satellite’s equilibria were determined as a result of the analysis of linearized equations of motion using Routh–Hurwitz criterion.
Introduction Analysis of the first exit time from the subset of states Markov queueing systems with fast service Single-server retrial queueing model Multiserver retrial queueing models Bibliography
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