Search

Let an n × n complex matrix A be such that I + A is invertible. The Cayley transform of A , denoted by C (A) , is defined as C (A) = (I + A) − 1 (I − A). Fallat and Tsatsomeros (2002) [5] and Mondal et al. (2024) [15] studied the Cayley transform of a matrix A in the context of P-matrices, H-matrices, M-matrices, totally positive matrices, positive definite matrices, almost skew-Hermitian matrices, and semipositive matrices. In this paper, the investigation of the Cayley transform is continued for Toeplitz matrices, circulant matrices, unipotent matrices, and dual matrices. An expression of the Cayley transform for dual matrices is established. It is shown that the Cayley transform of a dual symmetric matrix is always a dual symmetric matrix. The Cayley transform of a dual skew-symmetric matrix is discussed. The results are illustrated with examples. [ABSTRACT FROM AUTHOR]

Our focus in this study was on examining the convergence problem of a novel method, inspired by the Ulm-Chebyshev-like Cayley transform method, which was designed to solve the inverse eigenvalue problems (IEPs) with multiple eigenvalues. Compared with other existing methods, the proposed method has higher convergence order and/or requires less operations. Under the assumption that the relative generalized Jacobian matrices at a solution are nonsingular, the proposed method was proved to be convergent with cubic convergence. Experimental findings demonstrated the practicality and efficiency of the suggested approaches.

Let - A be the generator of a bounded C 0 -semigroup (e - t A) t ≥ 0 on a Hilbert space. First we study the long-time asymptotic behavior of the Cayley transform V ω (A) : = (A - ω I) (A + ω I) - 1 with ω > 0 . We give a decay estimate for ‖ V ω (A) n A - 1 ‖ when (e - t A) t ≥ 0 is polynomially stable. Considering the case where the parameter ω varies, we estimate ‖ (∏ k = 1 n V ω k (A)) A - 1 ‖ for exponentially stable C 0 -semigroups (e - t A) t ≥ 0 . Next we show that if the generator - A of the bounded C 0 -semigroup has a bounded inverse, then sup t ≥ 0 ‖ e - t A - 1 A - α ‖ < ∞ for all α > 0 . We also present an estimate for the rate of decay of ‖ e - t A - 1 A - 1 ‖ , assuming that (e - t A) t ≥ 0 is polynomially stable. To obtain these results, we use operator norm estimates offered by a functional calculus called the B -calculus. [ABSTRACT FROM AUTHOR]

We present a reformulation of optimization problems over the Stiefel manifold by using a Cayley-type transform, named the generalized left-localized Cayley transform, for the Stiefel manifold. The reformulated optimization problem is defined over a vector space, whereby we can apply directly powerful computational arts designed for optimization over a vector space. The proposed Cayley-type transform enjoys several key properties which are useful to (i) study relations between the original problem and the proposed problem; (ii) check the conditions to guarantee the global convergence of optimization algorithms. Numerical experiments demonstrate that the proposed algorithm outperforms the standard algorithms designed with a retraction on the Stiefel manifold. [ABSTRACT FROM AUTHOR]

We obtain the integrated error estimate of the Cayley transform method for solving the abstract Cauchy problem for the first-order differential equation with an unbounded operator coefficient in a Hilbert space. The estimate indicates that the method has a power rate of convergence, which automatically depends on the smoothness of the initial data (i.e., our method is a method without saturation of accuracy). Moreover, we substantiate that the estimate is unimprovable in the order of N (the discretization parameter N characterizes the number of summands in the partial sum for the approximate solution). [ABSTRACT FROM AUTHOR]

Aiming at the in-motion initial alignment problem of strapdown inertial navigation system (SINS), this paper proposed a precise direct in-motion alignment method based on attitude matrix representation and Cayley–Kalman filter on the special orthogonal group. There are two innovations in this paper. First, as an element of the special orthogonal group S O (3) , the error attitude matrix is used to construct an in-motion alignment model, and it is transformed into the vector space to represent the attitude error state. The error matrix was used as a state to decrease the influence of the state-dependent problem. Second, this method uses the Cayley transform to replace the Taylor expansion to derive the update equation, which is a no approximation mapping relation to assure accuracy when deriving the filtering algorithm. Simulation and experimental results demonstrate the proposed alignment method has advantages over the existing methods in alignment accuracy and alignment time and performs well in the in-motion alignment process of SINS. [ABSTRACT FROM AUTHOR]

Abstract Linear dynamic systems with impulsive effects are considered. For such a system we define a new impulsive exponential matrix. Necessary and sufficient conditions for exponential stability and boundedness have been established. The fundamental tool is an impulsive exponential matrix for exponential stability.

The author expounds or proves some results connected with Segal's chronometric theory. He gives short proofs of results about linear representation of the group of nondegenerate complex -matrices on the Minkowski space-time and on the universal covering of the Lie group of unitary -matrices, i.e., on the Einstein Universe, as well as about the Cayley transform of Lie algebras of Lie groups of unitary matrices into these groups. In comparison with the structure of conformal infinity for the Minkowski space, the structure of the set of unitary -matrices, which do not admit the Cayley transform, is found. Some problems are suggested. [ABSTRACT FROM AUTHOR]

Linear dynamic systems with impulsive effects are considered. For such a system we define a new impulsive exponential matrix. Necessary and sufficient conditions for exponential stability and boundedness have been established. The fundamental tool is an impulsive exponential matrix for exponential stability. [ABSTRACT FROM AUTHOR]

O'Dorney has proved that for any orthogonal matrix U , there exists a signature matrix D (a diagonal matrix with diagonal entries ±1) by which | s i j | ≤ 1 , where S = (s i j) = Q (U D) is the Cayley transform of UD. In this paper, we prove that the upper bound can be reduced to 2 − 1 by multiplying by a certain signed permutation matrix, instead of a signature matrix. [ABSTRACT FROM AUTHOR]

This paper proposes two innovative vector transport operators, leveraging the Cayley transform, for the generalized Stiefel manifold embedded with a non-standard metric. Specifically, it introduces the differentiated retraction and an approximation of the Cayley transform to the differentiated matrix exponential. These vector transports are demonstrated to satisfy the Ring–Wirth non-expansive condition under non-standard metrics, and one of them is also isometric. Building upon the novel vector transport operators, we extend the modified Polak–Ribière–Polyak (PRP) conjugate gradient method to the generalized Stiefel manifold. Under a non-monotone line search condition, we prove our algorithm globally converges to a stationary point. The efficiency of the proposed vector transport operators is empirically validated through numerical experiments involving generalized eigenvalue problems and canonical correlation analysis. [ABSTRACT FROM AUTHOR]

In this paper, we study the decay rate of the Cayley transform of the generator of a polynomially stable C 0 -semigroup. To estimate the decay rate of the Cayley transform, we develop an integral condition on resolvents for polynomial stability. Using this integral condition, we relate polynomial stability to Lyapunov equations. We also study robustness of polynomial stability for a certain class of structured perturbations. [ABSTRACT FROM AUTHOR]

In this paper, we study Cayley-transform-based gradient and conjugate gradient algorithms for optimization on Grassmann manifolds. We revisit the Cayley transform on Grassmann manifolds as a retraction in the framework of quotient manifolds constructed by Lie group actions and obtain an efficient formula for this retraction in low-rank cases. We also prove that this retraction is the restriction of the Cayley transform on Stiefel manifolds to horizontal spaces. To develop vector transports on Grassmann manifolds, we introduce a concept called induced vector transports on quotient manifolds. Based on this concept, three vector transports associated with the Cayley transform are obtained. The first vector transport is the traditional orthogonal projection onto horizontal spaces, whereas the other two vector transports are newly proposed herein. We show that one of the new vector transports satisfies the Ring–Wirth non-expansion condition and that the other is isometric. We also simplify the formulae of the new vector transports in low-rank cases. Riemannian gradient and conjugate gradient algorithms are established via the Cayley transform and the three abovementioned vector transports. Numerical experiments on two mean-of-subspaces problems demonstrate the effectiveness of the proposed algorithms. [ABSTRACT FROM AUTHOR]

The application of Cayley differential unitary space-time code in MIMO power line communication system was studied.The linear space of complex skew-Hermitian matrices was mapped to isotropic unitary matrix constellations represented by highly non-linear Stiefel manifolds,by which signals were transmitted through a block fading multiple input multiple output power line channel with in a differential mode.Finally,based on the unconstrained solution of the Cauchy random by which data streams were represented,the approximate linear maximum likelihood estimation of the transmitted signal was induced.Simulation results show that although the performance of the proposed scheme is slightly lower than that of the system based on orthogonal differential space-time code in terms of bit error rate and diversity gain,it can ensure high-speed and reliable communication at higher signal-to-noise ratios,and is suitable for multi-input multi-output power line communication systems requiring high-speed data transmission.

Minimizing a differentiable function restricted to the set of matrices with orthonormal columns finds applications in several fields of science and engineering. In this paper, we propose to solve this problem through a nonmonotone variation of the inexact restoration method consisting of two phases: restoration phase, aimed to improve feasibility, and minimization phase, aimed to decrease the function value in the tangent set of the constraints. For this, we give a suitable characterization of the tangent set of the orthogonality constraints, allowing us to (i) deal with the minimization phase efficiently and (ii) employ the Cayley transform to bring a point in the tangent set back to feasibility, leading to a SVD-free restoration phase. Based on previous global convergence results for the nonmonotone inexact restoration algorithm, it follows that any limit point of the sequence generated by the new algorithm is stationary. Moreover, numerical experiments on three different classes of the problem indicate that the proposed method is reliable and competitive with other recently developed methods. [ABSTRACT FROM AUTHOR]

The symplectic Stiefel manifold, denoted by Sp(2p, 2n), is the set of linear symplectic maps between the standard symplectic spaces ℝ2p and ℝ2n. When p = n, it reduces to the wellknown set of 2n \times 2n symplectic matrices. Optimization problems on Sp(2p, 2n) find applications in various areas, such as optics, quantum physics, numerical linear algebra, and model order reduction of dynamical systems. The purpose of this paper is to propose and analyze gradient-descent methods on Sp(2p, 2n), where the notion of gradient stems from a Riemannian metric. We consider a novel Riemannian metric on Sp(2p, 2n) akin to the canonical metric of the (standard) Stiefel manifold. In order to perform a feasible step along the antigradient, we develop two types of search strategies: one is based on quasi-geodesic curves and the other one on the symplectic Cayley transform. The resulting optimization algorithms are proved to converge globally to critical points of the objective function. Numerical experiments illustrate the efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

We consider the boundary value problems (BVPs) for linear second-order ODEs with a strongly positive operator coefficient in a Banach space. The solutions are given in the form of the infinite series by means of the Cayley transform of the operator, the Meixner type polynomials of the independent variable, the operator Green function, and the Fourier series representation for the right-hand side of the equation. The approximate solution of each problem is a partial sum of N (or expressed through N) summands. We prove the weighted error estimates depending on the discretization parameter N, the distance of the independent variable to the boundary points of the interval, and some smoothness properties of the input data. [ABSTRACT FROM AUTHOR]

We represent the solution u(t) of an initial value problem (IVP) for the first-order differential equation with an operator coefficient as a series using the Cayley transform of the corresponding operator coefficient and the Laguerre polynomials. In the case of a boundary value problem (BVP) for the second-order differential equation with an operator coefficient, we represent its solution using the Cayley transform and the Meixner-type polynomials. The approximate solution is the truncated sum of N (the discretization parameter) summands. We give the error estimate of these approximations depending on N and the distance of t to the initial point of the time interval or of the spatial argument x to the boundary of the spatial domain. [ABSTRACT FROM AUTHOR]

Summary: Since recent studies have shown that the Cayley transform method can be an effective iterative method for solving the inverse eigenvalue problem, in this work, we consider using an extension of it for solving a type of parameterized generalized inverse eigenvalue problem and prove its locally quadratic convergence. This type of inverse eigenvalue problem, which includes multiplicative and additive inverse eigenvalue problems, appears in many applications. Also, we consider the case where the given eigenvalues are multiple. In this case, we describe a modified problem that is not overdetermined and discuss the extension of the Cayley transform method for this modified problem. Finally, to demonstrate the effectiveness of these algorithms, we present some numerical examples to show that the proposed methods are practical and efficient. [ABSTRACT FROM AUTHOR]

A Cayley h-difference equation uses the forward-difference operator with step size h and then some proportion of the function value and the advanced function value. For such an equation with a complex constant coefficient, we establish that the equation exhibits instability along a certain circle, but is Hyers–Ulam stable inside and outside that circle; the circle becomes infinite and coincides with the vertical imaginary axis when the proportionality is equal. In the case of Hyers–Ulam stability, the best constant is found explicitly. The theory is also explained in terms of radial solutions, and an example is illustrated. [ABSTRACT FROM AUTHOR]

A boundary-value problem (BVP) for a second-order abstract differential equation with an operator coefficient in a Hilbert space is investigated. The exact solution is presented as an infinite series by means of the Cayley transform of the operator coefficient A and the polynomials of Meixner type in the independent variable x. The approximate solution is given by the truncated sum of the series with N addends. The error estimates (with the weighted function) depending not only on the discretization parameter N but also on the distance of the argument x to the boundary points of the segment are proved. The algorithm has a super-exponential rate of convergence. [ABSTRACT FROM AUTHOR]

Copyright of Cybernetics & Systems Analysis / Kibernetiki i Sistemnyj Analiz is the property of V.M. Glushkov Institute of Cybernetics of NAS of Ukraine and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Let ϕ be a subadditive weight on a C* -algebra A, and let Mϕ+ be the set of all elements x in A+ with ϕ(x) < +00. A seminorm ‖ • ‖ is introduced on the lineal Mϕsa = linRMϕ+, and a sufficient condition for the seminorm to be a norm is given. Let I be the unit of the algebra A, and let ϕ(I) = 1. Then, for every element x of Asa, the limit ρϕ(x) = limt→0+(ϕ(I + tx) - 1)/t exists and is finite. Properties of ρϕ are investigated, and examples of subadditive weights on C* -algebras are considered. On the basis of Lozinskii's 1958 results, specific subadditive weights on Mn(C) are considered. An estimate for the difference of Cayley transforms of Hermitian elements of a von Neumann algebra is obtained. [ABSTRACT FROM AUTHOR]

Following the recent work of Jiang and Lin (2020) [12] , we present more results (bounds) on Harnack type inequalities for matrices in terms of majorization (i.e., in partial products) of eigenvalues and singular values. We discuss and compare the bounds derived through different ways. Jiang and Lin's results imply Tung's version of Harnack's inequality (1964) [19] ; our results are stronger and more general than Jiang and Lin's. We also show some majorization inequalities concerning Cayley transforms. Some open problems on spectral norm and eigenvalues are proposed. [ABSTRACT FROM AUTHOR]

Let p be an odd prime. We show that each solution of the system of congruence equations x 1 2 + x 2 2 + x 3 2 + x 4 2 ≡ 0 (mod p) and x 1 + x 2 + x 3 + x 4 ≡ 0 (mod p) corresponds to precisely four solutions of the system of Diophantine equations x 1 2 + x 2 2 + x 3 2 + x 4 2 = p 2 and x 1 + x 2 + x 3 + x 4 = p that are pairwise orthogonal over Z , partially answering a conjecture proposed in Wang et al. [10]. The result was obtained by counting the number of solutions of both equations using Gaussian sum and modular forms, and the classical Cayley transformation. [ABSTRACT FROM AUTHOR]

The discretized Bethe-Salpeter eigenvalue problem arises in the Green's function evaluation in many body physics and quantum chemistry. Discretization leads to a matrix eigenvalue problem for H ∈ C2n-2n with a Hamiltonian-like structure. After an appropriate transformation of H to a standard symplectic form, the structure-preserving doubling algorithm, originally for algebraic Riccati equations, is extended for the discretized Bethe- Salpeter eigenvalue problem. Potential breakdowns of the algorithm, due to the ill condition or singularity of certain matrices, can be avoided with a double-Cayley transform or a three-recursion remedy. A detailed convergence analysis is conducted for the proposed algorithm, especially on the benign effects of the double-Cayley transform. Numerical results are presented to demonstrate the efficiency and the structure-preserving nature of the algorithm. [ABSTRACT FROM AUTHOR]

Copyright of Telecommunications Science is the property of Beijing Xintong Media Co., Ltd. and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

Let n be a nonempty, proper, convex subset of C . The n -maximal operators are defined as the operators having numerical ranges in n and are maximal with this property. Typical examples of these are the maximal symmetric (or accretive or dissipative) operators, the associated to some sesquilinear forms (for instance, to closed sectorial forms), and the generators of some strongly continuous semi-groups of bounded operators. In this paper the n -maximal operators are studied and some characterizations of these in terms of the resolvent set are given. [ABSTRACT FROM AUTHOR]

Most important computational problems nowadays are those related to processing of the large data sets and to numerical solution of the high-dimensional integral-differential equations. These problems arise in numerical modeling in quantum chemistry, material science, and multiparticle dynamics, as well as in machine learning, computer simulation of stochastic processes and many other applications related to big data analysis. Modern tensor numerical methods enable solution of the multidimensional partial differential equations (PDE) in ℝ d {\mathbb{R}^{d}} by reducing them to one-dimensional calculations. Thus, they allow to avoid the so-called "curse of dimensionality", i.e. exponential growth of the computational complexity in the dimension size d, in the course of numerical solution of high-dimensional problems. At present, both tensor numerical methods and multilinear algebra of big data continue to expand actively to further theoretical and applied research topics. This issue of CMAM is devoted to the recent developments in the theory of tensor numerical methods and their applications in scientific computing and data analysis. Current activities in this emerging field on the effective numerical modeling of temporal and stationary multidimensional PDEs and beyond are presented in the following ten articles, and some future trends are highlighted therein. [ABSTRACT FROM AUTHOR]

The Cayley transform maps the unit disk onto the upper half-plane, conformally and isometrically. In this paper, we generalize the Cayley transform in three-dimensional homogeneous geometries which are fiber bundles over the hyperbolic plane. Obtained generalizations are isometries between existing models in corresponding homogeneous geometries. Particularly, constructed isometry between two models of SL(2, R) geometry is nontrivial and enables comparison and transfer of known and even future results between these two models. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the spectral properties of a boundary value transmission problem generated by a dynamic equation on the union of two time scales. For such an analysis we assign a suitable dynamic operator which is in limit-circle case at infinity. We also show that this operator is a simple maximal dissipative operator. Constructing the inverse operator we obtain some information about the spectrum of the dissipative operator. Moreover, using the Cayley transform of the dissipative operator we pass to the contractive operator which is of the class C0. With the aid of the minimal function we obtain more information on the dissipative operator. Finally, we investigate other properties of the contraction such that multiplicity of the contraction, unitary colligation with basic operator and CMV matrix representation associated with the contraction. [ABSTRACT FROM AUTHOR]

Using the affine group and the Cayley transform from the unit disk D onto the upper half plane, we can turn D into a group, which we call the Poincaré unit disk. With this construction, D is a noncompact and nonunimodular Lie group. We characterize all infinite-dimensional, irreducible and unitary representations of D. By means of these representations, the Fourier transform on D is defined. The Plancherel theorem and hence the Fourier inversion formula can be given. Then pseudo-differential operators with operator-valued symbols, operator-valued Wigner transforms, and Weyl transforms on D are defined. [ABSTRACT FROM AUTHOR]

In this paper we define the quaternionic Cayley transformation of a densely defined, symmetric, quaternionic right linear operator and formulate a general theory of defect number in a right quaternionic Hilbert space. This study investigates the relation between the defect number and S-spectrum, and the properties of the Cayley transform in the quaternionic setting. [ABSTRACT FROM AUTHOR]

The Cayley transform for orthogonal groups is a well known construction with applications in real and complex analysis, linear algebra and computer science. In this work, we construct Cayley transforms on Stiefel manifolds. Applications to the Lusternik–Schnirelmann category and optimization problems are presented. [ABSTRACT FROM AUTHOR]