Search

We have considered a generalized Sylvester-transpose matrix equation $ AXB + CX^TD = E, $ where $ A, B, C, D, $ and $ E $ are given rectangular matrices over a generalized quaternion skew-field, and $ X $ is an unknown matrix. We have applied certain vectorizations and real representations to transform the matrix equation into a matrix equation over the real numbers. Thus, we have investigated a solvability condition, general exact/least-squares solutions, minimal-norm solutions, and the exact/least-squares solution closest to a given matrix. The main equation included the equation $ AXB = E $ and the Sylvester-transpose equation. Our results also covered such matrix equations over the quaternions, and quaternionic linear systems.

We characterized weighted spectral geometric means (SGM) of positive definite matrices in terms of certain matrix equations involving metric geometric means (MGM) $ \sharp $ and semi-tensor products $ \ltimes $. Indeed, for each real number $ t $ and two positive definite matrices $ A $ and $ B $ of arbitrary sizes, the $ t $-weighted SGM $ A \, \diamondsuit_t \, B $ of $ A $ and $ B $ is a unique positive solution $ X $ of the equation $ A^{-1}\,\sharp\, X \; = \; (A^{-1}\,\sharp\, B)^t. $ We then established fundamental properties of the weighted SGMs based on MGMs. In addition, $ (A \, \diamondsuit_{1/2} \, B)^2 $ is positively similar to $ A \ltimes B $ and, thus, they have the same spectrum. Furthermore, we showed that certain equations concerning weighted SGMs and MGMs of positive definite matrices have a unique solution in terms of weighted SGMs. Our results included the classical weighted SGMs of matrices as a special case.

We extend the notion of classical metric geometric mean (MGM) for positive definite matrices of the same dimension to those of arbitrary dimensions, so that usual matrix products are replaced by semi-tensor products. When the weights are arbitrary real numbers, the weighted MGMs possess not only nice properties as in the classical case, but also affine change of parameters, exponential law, and cancellability. Moreover, when the weights belong to the unit interval, the weighted MGM has remarkable properties, namely, monotonicity and continuity from above. Then we apply a continuity argument to extend the weighted MGM to positive semidefinite matrices, here the weights belong to the unit interval. It turns out that this matrix mean posses rich algebraic, order, and analytic properties, such as, monotonicity, continuity from above, congruent invariance, permutation invariance, affine change of parameters, and exponential law. Furthermore, we investigate certain equations concerning weighted MGMs of positive definite matrices. It turns out that such equations are always uniquely solvable with explicit solutions. The notion of MGMs can be applied to solve certain symmetric word equations in two letters.

We investigate the Riccati matrix equation $ X A^{-1} X = B $ in which the conventional matrix products are generalized to the semi-tensor products $ \ltimes $. When $ A $ and $ B $ are positive definite matrices satisfying the factor-dimension condition, this equation has a unique positive definite solution, which is defined to be the metric geometric mean of $ A $ and $ B $. We show that this geometric mean is the maximum solution of the Riccati inequality. We then extend the notion of the metric geometric mean to positive semidefinite matrices by a continuity argument and investigate its algebraic properties, order properties and analytic properties. Moreover, we establish some equations and inequalities of metric geometric means for matrices involving cancellability, positive linear map and concavity. Our results generalize the conventional metric geometric means of matrices.

Consider a linear system Ax=b where the coefficient matrix A is rectangular and of full-column rank. We propose an iterative algorithm for solving this linear system, based on gradient-descent optimization technique, aiming to produce a sequence of well-approximate least-squares solutions. Here, we consider least-squares solutions in a full generality, that is, we measure any related error through an arbitrary vector norm induced from weighted positive definite matrices W. It turns out that when the system has a unique solution, the proposed algorithm produces approximated solutions converging to the unique solution. When the system is inconsistent, the sequence of residual norms converges to the weighted least-squares error. Our work includes the usual least-squares solution when W=I. Numerical experiments are performed to validate the capability of the algorithm. Moreover, the performance of this algorithm is better than that of recent gradient-based iterative algorithms in both iteration numbers and computational time.

We consider the two-dimensional space-time fractional differential equation with the Caputo's time derivative and the Riemann-Liouville space derivatives on bounded domains. The equation is subjected to the zero Dirichlet boundary condition and the zero initial condition. We discretize the equation by finite difference schemes based on Grünwald-Letnikov approximation. Then we linearize the discretized equations into a sparse linear system. To solve such linear system, we propose a gradient-descent iterative algorithm with a sequence of optimal convergence factor aiming to minimize the error occurring at each iteration. The convergence analysis guarantees the capability of the algorithm as long as the coefficient matrix is invertible. In addition, the convergence rate and error estimates are provided. Numerical experiments demonstrate the efficiency, the accuracy and the performance of the proposed algorithm.

We develop an effective algorithm to find a well-approximate solution of a generalized Sylvester-transpose matrix equation where all coefficient matrices and an unknown matrix are rectangular. The algorithm aims to construct a finite sequence of approximated solutions from any given initial matrix. It turns out that the associated residual matrices are orthogonal, and thus, the desire solution comes out in the final step with a satisfactory error. We provide numerical experiments to show the capability and performance of the algorithm.

Consider a generalized Sylvester-transpose matrix equation with rectangular coefficient matrices. Based on gradients and hierarchical identification principle, we derive an iterative algorithm to produce a sequence of approximated solutions with a reasonable stopping rule concerning a relative norm-error. A convergence analysis via Banach fixed-point theorem reveals the sequence converges to a unique solution of the matrix equation for any given initial matrix if and only if the convergence factor is chosen appropriately in a certain range. The performance of algorithm is theoretically analysed through the convergence rate and error estimations. The optimal convergence factor is chosen to attain the fastest asymptotic behaviour. Finally, numerical experiments are provided to illustrate the capability and efficiency of the proposed algorithm, compared to recent gradient-based iterative algorithms.

Abstract This paper proposes an effective gradient-descent iterative algorithm for solving a generalized Sylvester-transpose equation with rectangular matrix coefficients. The algorithm is applicable for the equation and its interesting special cases when the associated matrix has full column-rank. The main idea of the algorithm is to have a minimum error at each iteration. The algorithm produces a sequence of approximated solutions converging to either the unique solution, or the unique least-squares solution when the problem has no solution. The convergence analysis points out that the algorithm converges fast for a small condition number of the associated matrix. Numerical examples demonstrate the efficiency and effectiveness of the algorithm compared to renowned and recent iterative methods.

Abstract We derive an iterative procedure for solving a generalized Sylvester matrix equation A X B + C X D = E $AXB+CXD = E$ , where A , B , C , D , E $A,B,C,D,E$ are conforming rectangular matrices. Our algorithm is based on gradients and hierarchical identification principle. We convert the matrix iteration process to a first-order linear difference vector equation with matrix coefficient. The Banach contraction principle reveals that the sequence of approximated solutions converges to the exact solution for any initial matrix if and only if the convergence factor belongs to an open interval. The contraction principle also gives the convergence rate and the error analysis, governed by the spectral radius of the associated iteration matrix. We obtain the fastest convergence factor so that the spectral radius of the iteration matrix is minimized. In particular, we obtain iterative algorithms for the matrix equation A X B = C $AXB=C$ , the Sylvester equation, and the Kalman–Yakubovich equation. We give numerical experiments of the proposed algorithm to illustrate its applicability, effectiveness, and efficiency.

Abstract In this paper, we introduce a new iterative algorithm for solving a generalized Sylvester matrix equation of the form ∑ t = 1 p A t X B t = C $\sum_{t=1}^{p}A_{t}XB_{t}=C$ which includes a class of linear matrix equations. The objective of the algorithm is to minimize an error at each iteration by the idea of gradient-descent. We show that the proposed algorithm is widely applied to any problems with any initial matrices as long as such problem has a unique solution. The convergence rate and error estimates are given in terms of the condition number of the associated iteration matrix. Furthermore, we apply the proposed algorithm to sparse systems arising from discretizations of the one-dimensional heat equation and the two-dimensional Poisson’s equation. Numerical simulations illustrate the capability and effectiveness of the proposed algorithm comparing to well-known methods and recent methods.

Abstract We introduce an effective iterative method for solving rectangular linear systems, based on gradients along with the steepest descent optimization. We show that the proposed method is applicable with any initial vectors as long as the coefficient matrix is of full column rank. Convergence analysis produces error estimates and the asymptotic convergence rate of the algorithm, which is governed by the term 1 − κ − 2 $\sqrt {1-\kappa^{-2}}$ , where κ is the condition number of the coefficient matrix. Moreover, we apply the proposed method to a sparse linear system arising from a discretization of the one-dimensional Poisson equation. Numerical simulations illustrate the capability and effectiveness of the proposed method in comparison to the well-known and recent methods.

We formulate a mathematical model for the classical Chua’s circuit with two nonlinear resistors in terms of a system of nonlinear ordinary differential equations. The existence of two nonlinear resistors implies that the system has three equilibrium points. The behaviour of the trajectory in a neighbourhood of each equilibrium point depends on the eigenvalues of the system. The eigenvalues can be obtained from a cubic polynomial equation. It turns out that all possible solutions of the cubic equation lead to six types of equilibrium points, namely, stable node, unstable node, saddle node, stable focus node, unstable focus node, saddle focus node. The chaotic behaviour of the circuit occurs when the equilibrium point is a stable focus node or a saddle focus node. The hidden attractor of our Chua’s system is localized through a suitable initial point.

We derive a conjugate-gradient type algorithm to produce approximate least-squares (LS) solutions for an inconsistent generalized Sylvester-transpose matrix equation. The algorithm is always applicable for any given initial matrix and will arrive at an LS solution within finite steps. When the matrix equation has many LS solutions, the algorithm can search for the one with minimal Frobenius-norm. Moreover, given a matrix Y, the algorithm can find a unique LS solution closest to Y. Numerical experiments show the relevance of the algorithm for square/non-square dense/sparse matrices of medium/large sizes. The algorithm works well in both the number of iterations and the computation time, compared to the direct Kronecker linearization and well-known iterative methods.

We introduce the Tracy-Singh sum for operators on a Hilbert space, generalizing both the Tracy-Singh sum for matrices and the tensor sum for operators. The Tracy-Singh sum is shown to be compatible with algebraic operations and order relations. Then we establish a binomial theorem involving Tracy-Singh sums, and its consequences. We also investigate continuity, convergence, and norm bounds for Tracy-Singh sums. Moreover, we derive operator identities involving Tracy-Singh sums and certain operator functions defined by power series.

This paper investigates the Sylvester-transpose matrix equation A⋉X+XT⋉B=C, where all mentioned matrices are over an arbitrary field. Here, ⋉ is the semi-tensor product, which is a generalization of the usual matrix product defined for matrices of arbitrary dimensions. For matrices of compatible dimensions, we investigate criteria for the equation to have a solution, a unique solution, or infinitely many solutions. These conditions rely on ranks and linear dependence. Moreover, we find suitable matrix partitions so that the matrix equation can be transformed into a linear system involving the usual matrix product. Our work includes the studies of the equation A⋉X=C, the equation X⋉B=C, and the classical Sylvester-transpose matrix equation.

We generalize the notions of Khatri-Rao sums for matrices and tensor sums for Hilbert space operators to KhatriRao sums for Hilbert space operators. This kind of operator sum is compatible with algebraic operations and order relations. We investigate its analytic properties, including continuity, convergence, and norm bounds. We also discuss the role of selection operator that relates Khatri-Rao sums to Tracy-Singh sums and Khatri-Rao products. Binomial theorem involving Khatri-Rao sums and its consequences are then established.

We propose a new iterative method for solving a generalized Sylvester matrix equation A1XA2+A3XA4=E with given square matrices A1,A2,A3,A4 and an unknown rectangular matrix X. The method aims to construct a sequence of approximated solutions converging to the exact solution, no matter the initial value is. We decompose the coefficient matrices to be the sum of its diagonal part and others. The recursive formula for the iteration is derived from the gradients of quadratic norm-error functions, together with the hierarchical identification principle. We find equivalent conditions on a convergent factor, relied on eigenvalues of the associated iteration matrix, so that the method is applicable as desired. The convergence rate and error estimation of the method are governed by the spectral norm of the related iteration matrix. Furthermore, we illustrate numerical examples of the proposed method to show its capability and efficacy, compared to recent gradient-based iterative methods.

We introduce a gradient iterative scheme with an optimal convergent factor for solving a generalized Sylvester matrix equation ∑i=1pAiXBi=F, where Ai,Bi and F are conformable rectangular matrices. The iterative scheme is derived from the gradients of the squared norm-errors of the associated subsystems for the equation. The convergence analysis reveals that the sequence of approximated solutions converge to the exact solution for any initial value if and only if the convergent factor is chosen properly in terms of the spectral radius of the associated iteration matrix. We also discuss the convergent rate and error estimations. Moreover, we determine the fastest convergent factor so that the associated iteration matrix has the smallest spectral radius. Furthermore, we provide numerical examples to illustrate the capability and efficiency of this method. Finally, we apply the proposed scheme to discretized equations for boundary value problems involving convection and diffusion.

This paper investigates the chaotic behavior of a modified jerk circuit with Chua’s diode. The Chua’s diode considered here is a nonlinear resistor having a symmetric piecewise linear voltage-current characteristic. To describe the system, we apply fundamental laws in electrical circuit theory to formulate a mathematical model in terms of a third-order (jerk) nonlinear differential equation, or equivalently, a system of three first-order differential equations. The analysis shows that this system has three collinear equilibrium points. The time waveform and the trajectories about each equilibrium point depend on its associated eigenvalues. We prove that all three equilibrium points are of type saddle focus, meaning that the trajectory of (x(t),y(t)) diverges in a spiral form but z(t) converges to the equilibrium point for any initial point (x(0),y(0),z(0)). Numerical simulation illustrates that the oscillations are dense, have no period, are highly sensitive to initial conditions, and have a chaotic hidden attractor.

Abstract We present a number of integral inequalities involving tensor products of continuous fields of bounded linear operators, positive linear maps, and operator means. In particular, the Kantorovich, Grüss, and reverse Hölder-McCarthy integral inequalities are obtained as special cases.

In this paper, we survey the role of mathematics in electrical network connections. We discuss the behavior of current flows, voltages and impedances mainly for series-parallel networks. In both one-port and multiport electrical networks, the currents through these networks are governed by Maxwell’s power principle. The joint impedances of the networks, given in terms of series sum and parallel sum, satisfy the series-parallel inequality. An abstract idea can be formulated in functional analysis aspect in which any network connection is viewed to be a binary operation for positive operators satisfying certain algebraic, order and topological properties.

We consider bounded continuous fields of self-adjoint operators which are parametrized by a locally compact Hausdorff space Ω equipped with a finite Radon measure μ . Under certain assumptions on synchronous Khatri−Rao property of the fields of operators, we obtain Chebyshev-type inequalities concerning Khatri−Rao products. We also establish Chebyshev-type inequalities involving Khatri−Rao products and weighted Pythagorean means under certain assumptions of synchronous monotone property of the fields of operators. The Pythagorean means considered here are three classical symmetric means: the geometric mean, the arithmetic mean, and the harmonic mean. Moreover, we derive the Chebyshev−Grüss integral inequality via oscillations when μ is a probability Radon measure. These integral inequalities can be reduced to discrete inequalities by setting Ω to be a finite space equipped with the counting measure. Our results provide analog results for matrices and integrable functions. Furthermore, our results include the results for tensor products of operators, and Khatri−Rao/Kronecker/Hadamard products of matrices, which have been not investigated in the literature.

We investigate a descriptor system of coupled generalized Sylvester matrix fractional differential equations in both non-homogeneous and homogeneous cases. All fractional derivatives considered here are taken in Caputo’s sense. We explain a 4-step procedure to solve the descriptor system, consisting of vectorization, a matrix canonical form concerning ranks, and matrix partitioning. The procedure aims to reduce the descriptor system to a descriptor system of fractional differential equations. We also impose a condition on coefficient matrices, related to the symmetry of the solution for descriptor systems. It follows that an explicit form of its general solution is given in terms of matrix power series concerning Mittag–Leffler functions. The main system includes certain systems of coupled matrix/vector differential equations, and single matrix differential equations as special cases. In particular, we obtain an alternative procedure to solve linear continuous-time descriptor systems via a matrix canonical form.

In this paper, we establish several integral inequalities of Chebyshev type for bounded continuous fields of Hermitian operators concerning Tracy-Singh products and weighted Pythagorean means. The weighted Pythagorean means considered here are parametrization versions of three symmetric means: the arithmetic mean, the geometric mean, and the harmonic mean. Every continuous field considered here is parametrized by a locally compact Hausdorff space equipped with a finite Radon measure. Tracy-Singh product versions of the Chebyshev-Grüss inequality via oscillations are also obtained. Such integral inequalities reduce to discrete inequalities when the space is a finite space equipped with the counting measure. Moreover, our results include Chebyshev-type inequalities for tensor product of operators and Tracy-Singh/Kronecker products of matrices.

We introduce the notion of Khatri-Rao product for operator matrices acting on the direct sum of Hilbert spaces. This notion generalizes the tensor product and Hadamard product of operators and the Khatri-Rao product of matrices. We investigate algebraic properties, positivity, and monotonicity of the Khatri-Rao product. Moreover, there is a unital positive linear map taking Tracy-Singh products to Khatri-Rao products via an isometry.

An operator mean is a binary operation assigned to each pair of positive operators satisfying monotonicity, continuity from above, the transformer inequality, and the fixed-point property. It is well known that there are one-to-one correspondences between operator means, operator monotone functions, and Borel measures. In this paper, we provide various characterizations for the concepts of positivity, betweenness, and strictness of operator means in terms of operator inequalities, operator monotone functions, Borel measures, and certain operator equations.

We show how to construct new Finsler metrics, in two and three dimensions, whose indicatrices are pedal curves or pedal surfaces of some other curves or surfaces, respectively. These Finsler metrics are generalizations of the famous slope metric, also called Matsumoto metric. [ABSTRACT FROM AUTHOR]