Non-square Matrix Research Articles

Diagonal Scalings for the Eigenstructure of Arbitrary Pencils

In this paper we show how to construct diagonal scalings for arbitrary matrix pencils $\lambda B-A$, in which both $A$ and $B$ are complex matrices (square or nonsquare). The goal of such diagonal scalings is to balance in some sense the row and column norms of the pencil. We see that the problem of scaling a matrix pencil is equivalent to the problem of scaling the row and column sums of a particular nonnegative matrix. However, it is known that there exist square and nonsquare nonnegative matrices that can not be scaled arbitrarily. To address this issue, we consider an approximate embedded problem, in which the corresponding nonnegative matrix is square and can always be scaled. The new scaling methods are then based on the Sinkhorn-Knopp algorithm for scaling a square nonnegative matrix with total support to be doubly stochastic or on a variant of it. In addition, using results of U. G. Rothblum and H. Schneider (1989), we give simple sufficient conditions on the zero pattern for the existence of diagonal scalings of square nonnegative matrices to have any prescribed common vector for the row and column sums. We illustrate numerically that the new scaling techniques for pencils improve the accuracy of the computation of their eigenvalues.

Quantum Linear System Algorithm for General Matrices in System Identification

Solving linear systems of equations is one of the most common and basic problems in classical identification systems. Given a coefficient matrix A and a vector b, the ultimate task is to find the solution x such that . Based on the technique of the singular value estimation, the paper proposes a modified quantum scheme to obtain the quantum state corresponding to the solution of the linear system of equations in poly time for a general dimensional A, which is superior to existing quantum algorithms, where is the condition number, r is the rank of matrix A and is the precision parameter. Meanwhile, we also design a quantum circuit for the homogeneous linear equations and achieve an exponential improvement. The coefficient matrix A in our scheme is a sparsity-independent and non-square matrix, which can be applied in more general situations. Our research provides a universal quantum linear system solver and can enrich the research scope of quantum computation.

Hybridization of Manta-Ray Foraging Optimization Algorithm with Pseudo Parameter-Based Genetic Algorithm for Dealing Optimization Problems and Unit Commitment Problem

The manta ray foraging optimization algorithm (MRFO) is one of the promised meta-heuristic optimization algorithms. However, it can stick to a local minimum, consuming iterations without reaching the optimum solution. So, this paper proposes a hybridization between MRFO, and the genetic algorithm (GA) based on a pseudo parameter; where the GA can help MRFO to escape from falling into the local minimum. It is called a pseudo genetic algorithm with manta-ray foraging optimization (PGA-MRFO). The proposed algorithm is not a classical hybridization between MRFO and GA, wherein the classical hybridization consumes time in the search process as each algorithm is applied to all system variables. In addition, the classical hybridization results in an extended search algorithm, especially in systems with many variables. The PGA-MRFO hybridizes the pseudo-parameter-based GA and the MRFO algorithm to produce a more efficient algorithm that combines the advantages of both algorithms without getting stuck in a local minimum or taking a long time in the calculations. The pseudo parameter enables the GA to be applied to a specific number of variables and not to all system variables leading to reduce the computation time and burden. Also, the proposed algorithm used an approximation for the gradient of the objective function, which leads to dispensing derivatives calculations. Besides, PGA-MRFO depends on the pseudo inverse of non-square matrices, which saves calculations time; where the dependence on the pseudo inverse gives the algorithm more flexibility to deal with square and non-square systems. The proposed algorithm will be tested on the test functions that the main MRFO failed to find their optimum solution to prove its capability and efficiency. In addition, it will be applied to solve the unit commitment (UC) problem as one of the vital power system problems to show the validity of the proposed algorithm in practical applications. Finally, several analyses will be applied to the proposed algorithm to illustrate its effectiveness and reliability.

Toeplitz momentary symbols: definition, results, and limitations in the spectral analysis of structured matrices

A powerful tool for analyzing and approximating the singular values and eigenvalues of structured matrices is the theory of Generalized Locally Toeplitz (GLT) sequences. By the GLT theory one can derive a function, called the symbol, which describes the singular value or the eigenvalue distribution of the sequence, the latter under precise assumptions. However, for small values of the matrix-size of the considered sequence, the approximations may not be as good as it is desirable, since in the construction of the GLT symbol one disregards small norm and low-rank perturbations. On the other hand, Local Fourier Analysis (LFA) can be used to construct polynomial symbols in a similar manner for discretizations, where the geometric information is present, but the small norm perturbations are retained.The main focus of this paper is the introduction of the concept of sequence of “Toeplitz momentary symbols”, associated with a given sequence of truncated Toeplitz-like matrices. We construct the symbol in the same way as in the GLT theory, but we keep the information of the small norm contributions. The low-rank contributions are still disregarded, and we give an idea on the reason why this is negligible in certain cases and why it is not in other cases, being aware that in presence of high nonnormality the same low-rank perturbation can produce a dramatic change in the eigenvalue distribution. Moreover, a difference with respect to the LFA symbols is that GLT symbols and Toeplitz momentary symbols are more general - just Lebesgue measurable - and are applicable to a larger class of matrices, while in the LFA setting only trigonometric polynomials are considered and more specifically those related to the approximation stencils. We show the applicability of the approach which leads to higher accuracy in some cases, when approximating the singular values and eigenvalues of Toeplitz-like matrices using Toeplitz momentary symbols, compared with the GLT symbol. Finally, since for many applications and their analysis it is often necessary to consider non-square Toeplitz matrices, we formalize and provide some useful definitions, applicable for non-square Toeplitz momentary symbols.

An Effective Non-Square Matrix Converter Based Approach for Active Power Control of Multiple DGs in Microgrids: Experimental Implementation

In this paper, a new modulation strategy based on the carrier-based switching strategy for the non-square direct matrix converters (MC) is proposed to control the active power of distributed generation (DG) units. In this strategy, the active power of DGs is controlled by the central input current control of the non-square direct MC independent from the voltage and frequency. Conventionally, each DG has a converter, and for supplying a load with N number of DGs, N number of converters are needed and each converter has its own modulation switching and control strategy to control the power output of each DG. Needless to say, in a microgrid with N number of DGs, the control strategy of each converter has more complex structure than that of a microgrid with one converter, and surely the former strategy entails more volume and price. Using the proposed converter in this paper, it is possible to supply a load with N number of DGs through one converter. Also, the power outputs of all DGs are controlled by a central control strategy. The proposed central control strategy is described and simulated for a typical microgrid. Experimental and simulation results validate the effectiveness of the proposed converter and the proposed strategy. The results demonstrate the applicability and efficiency of the system and verify the theoretical analysis.

Полиномиальный метод синтеза регуляторов по задающему и возмущающим воздействиям

Linear controlled objects with one input and one output (single input - single output, SISO), and objects with multiple inputs and multiple outputs (multi-input – multi-output, MIMO) have different formalized controller synthesis algorithms. At the same time, objects with an unequal number of inputs and outputs, in many cases, are built by the developer intuitively, when changing the existing calculation algorithms for each control object, therefore, the development of a formalized calculation algorithm for this type of objects is relevant. Within the framework of this work, it is proposed to extend the synthesis technique for multichannel objects, which is the polynomial synthesis technique, to objects with a smaller number of inputs compared to the number of outputs, namely, to objects with one input and several outputs (single input – multi-output, SIMO). The reasoning developed in the work is an example of calculating an electromechanical tension control system in the material transportation zone of the production line, which has one input – the voltage supplied to the electric motor and four outputs–- the armature current, the rotation speed of the electric motor shaft, the rotation speed of the roll, the tension in the zone under consideration and the elastic moment. The tension in the considered zone is an adjustable coordinate. The use of the polynomial synthesis method for objects with a non-square matrix function made it possible to place the poles of a closed system in a given position, and the transfer function does not contain zeros according to the assignment. It was also possible to set the disturbance-stimulated zeros of the closed system in such a way that a second-order astatism is obtained.

Optimal singular value shrinkage for operator norm loss: Extending to non-square matrices

We correct a formula of Gavish and Donoho for singular value shrinkage with operator norm loss for non-square matrices. We also observe that in the classical regime, optimal shrinkage for any Schatten loss converges to the best linear predictor.

A Generalized CUR Decomposition for Matrix Pairs

We propose a generalized CUR (GCUR) decomposition for matrix pairs $(A, B)$. Given matrices $A$ and $B$ with the same number of columns, such a decomposition provides low-rank approximations of both matrices simultaneously, in terms of some of their rows and columns. We obtain the indices for selecting the subset of rows and columns of the original matrices using the discrete empirical interpolation method (DEIM) on the generalized singular vectors. When $B$ is square and nonsingular, there are close connections between the GCUR of $(A, B)$ and the DEIM-induced CUR of $AB^{-1}$. When $B$ is the identity, the GCUR decomposition of $A$ coincides with the DEIM-induced CUR decomposition of $A$. We also show a similar connection between the GCUR of $(A, B)$ and the CUR of $AB^+$ for a nonsquare but full-rank matrix $B$, where $B^+$ denotes the Moore--Penrose pseudoinverse of $B$. While a CUR decomposition acts on one data set, a GCUR factorization jointly decomposes two data sets. The algorithm may be suitable for applications where one is interested in extracting the most discriminative features from one data set relative to another data set. In numerical experiments, we demonstrate the advantages of the new method over the standard CUR approximation; for recovering data perturbed with colored noise and subgroup discovery.

Extended dissipativity and dynamical output feedback control for interval type-2 singular semi-Markovian jump fuzzy systems

This paper considers the problems of stochastic admissibility and extended dissipativity analysis, as well as dynamical output feedback controller design for interval type-2 singular semi-Markovian jump fuzzy systems. By utilising the partition method and the matrix decomposition technique, an improved approach of handling the time-varying transition rates is proposed. Based on this approach, a sufficient condition with less conservatism is established to guarantee that the closed-loop system is regular, impulse-free, exponentially stable in mean square and extended dissipative. Determining the parametric matrices of the controller via the obtained sufficient condition is difficult, since some parametric matrices factor between two nonsquare matrices. A dynamical output feedback control scheme is proposed by utilising a vital result which is proposed inspired by Projection lemma. It is worth noting that no conservatism is introduced in controller design due to the sufficiency and necessity of this result. Finally, simulation examples are given to show the effectiveness of the proposed scheme.

Adaptive robust multivariable control of noninvertible memoryless systems with bounded disturbances: a generalization

The discrete-time robust adaptive control for some classes of the uncertain multivariable memoryless (static) systems in the presence of unmeasurable bounded disturbances, whose bounds are assumed to be known is addressed. The systems where the number of the control inputs do not exceed the number of their outputs are considered. The main feature of plants to be controlled is that their gain matrices are noninvertible. The assumption that the elements of these matrices are unknown a priori but there is information about possible bounds on these elements. The problem stated and solvesd here is to design a feedback controller to be capable to cope with the noninvertibility of the gain matrices and also with the parametric uncertainty in order to reject the external disturbances and to ensure the boun­dedness of all the control and output system signals. To solve the problem above mentioned, the robust adaptive approach together with the so-called pseudoinverse or inverse model-based concept is used. Three different cases are studied. In the first case, the robust adaptive controller applicable to the uncertain plant with the square singular gain matrix is designed. The robust method employing the pseudoinverse model-based controllers whose parameters are estimated via a standard recursive adaptation procedure is proposed in the second case to deal with the unknown nonsquare gain matrices having the full rank. The approach proposed in first case is extended to the third case dealing with the control of the unknown plants the gain matrices of which represent the nonsquare matrices of not full rank. Asymptotic properties of the robustly-adaptive controllers proposed in this paper are established. Results of numerical examples given to support the theoretic study.

INVERS TERGENERALISASI MOORE PENROSE

The generalized inverse is a concept for determining the inverse of a singular matrix and and matrix which has the characteristic of the inverse matrix. There are several types of generalized inverse, one of which is the Moore-Penrose inverse. The matrix is called Moore Penrose inverse of a matrix if it satisfies the four penrose equations and is denoted by . Furthermore, if the matrix satisfies only the first two equations of the Moore-Penrose inverse and , then is called the group inverse of and is denoted by . The purpose of this research was to determine the group inverse of a non-diagonalizable square matrix using Jordan’s canonical form and Moore Penrose’s inverse of a singular matrix, also a non-square matrix using the Singular Value Decomposition (SVD) method. The results of this study are the sufficient condition for a matrix to have a group inverse, i.e., a matrix has an index of 1 if and only if the product of two matrices forming is a full rank factorization and is invertible. Whereas for a singular matrix and a non-square , the Moore-Penrose inverse can be determined using Singular Value Decomposition (SVD). Keywords: generalized matrix inverse, Moore Penrose inverse, group inverse, Jordan canonical form, Singular Value Decomposition.

From Penrose Equations to Zhang Neural Network, Getz–Marsden Dynamic System, and DDD (Direct Derivative Dynamics) Using Substitution Technique

The time-variant matrix inversion (TVMI) problem solving is the hotspot of current research because of its frequent appearance and application in scientific research and industrial production. The generalized inverse problem of singular square matrix and nonsquare matrix can be related to Penrose equations (PEs). The PEs implicitly define the generalized inverse of a known matrix, which is of fundamental theoretical significance. Therefore, the in-depth study of PEs might enlighten problem solving of TVMI in a foreseeable way. For the first time, we construct three different matrix error-monitoring functions based on PEs and propose the corresponding models for TVMI problem solving by using the substitution technique and ZNN design formula. In order to facilitate computer simulation, the obtained continuous-time models are discretized by using ZTD (Zhang time discretization) formulas. Furthermore, the feasibility and effectiveness of the novel Zhang neural network (ZNN) multiple-multiplication model for matrix inverse (ZMMMI) and the PEs-based Getz–Marsden dynamic system (PGMDS) model in solving the problem of TVMI are investigated and shown via theoretical derivation and computer simulation. Computer experiment results also illustrate that the direct derivative dynamics model for TVMI is less effective and feasible.

Полиномиальный метод синтеза регуляторов для частного случая многоканальных объектов с одной входной переменной и несколькими выходными

Currently, an urgent task in control theory is the synthesis of regulators for objects with a smaller number of input values compared to output ones, such objects are described by matrix transfer functions of a non-square shape. A particular case of a multichannel object with one input variable and two / three / four output variables is considered; the matrix transfer function of such an object has not a square shape, but one column and two / three / four rows. To calculate the controllers, a polynomial synthesis technique is used, which consists in using a polynomial matrix description of a closed-loop control system. A feature of this approach is the ability to write the characteristic matrix of a closed multichannel system through the polynomial matrices of the object and the controller in the form of a matrix Diophantine equation. By solving the Diophantine equation, the desired poles of the matrix characteristic polynomial of the closed system are set. There are many options for solving the Diophantine equation and one of them is to represent the polynomial matrix Diophantine equation as a system of linear algebraic equations in matrix form, where the matrix of the system is the Sylvester matrix. The choice of the order of the polynomial matrix controller and the order of the characteristic matrix is carried out on the basis of the theorem given in the works of Chi-Tsong Chen, which always holds for controlled objects. If the minimum order of the controller is chosen in accordance with this theorem, and the Sylvester matrix has not full rank, then this means that there are more unknown elements in the system of linear algebraic equations than there are equations. In this case, the solution corresponding to the selected basic minor has free parameters, which are the parameters of the regulators. Free parameters of regulators can be set arbitrarily, which is used to set or exclude some zeros in a closed system. Thus, using various examples of objects with a non-square matrix transfer function, a polynomial synthesis technique is illustrated, which allows not only specifying the poles of a closed system, but also some zeros, which is a significant advantage, especially when synthesizing controllers for multichannel objects.

Enhanced energy harvesting from rooftop PV array using Block Swap algorithm

Partial Shading conditions arising from tall buildings, upcoming infrastructural setup, trees, passing clouds, and other factors adversely affect the Photo-Voltaic (PV) array output power and overall efficiency. This paper proposes a field implementable PV array reconfiguration technique named Block Swap-inspired Fixed Column (BSFC) reconfiguration. The proposed reconfiguration technique makes one-time connections (at the time of PV array installation). It eliminates the need to physically relocate the PV panel as is propounded by various static and dynamic reconfiguration techniques reported in the literature. The physical relocation of PV panels is not a viable real-time solution for shade dispersion due to the shading patterns' stochastic nature. The proposed technique works well for both square (m × m) and non-square matrix (m × n) dimensions of the PV array and provides better results for all shading patterns witnessed over the length of the day. The effectiveness and feasibility of the proposed reconfiguration technique (BSFC) are tested on a solar PV rooftop array installed at IIT Jammu Campus, and the simulation is done on the MATLAB platform. Simulation results confirm the potential and effectiveness of BSFC after comparison with other reconfiguration methods like Series-Parallel (SP), Total Cross Tied (TCT), Odd-Even (OE), Odd-Even Prime (OEP). The proposed technique provides enhanced power output while utilizing the least length of the connecting dc cables and the least electrical displacement w.r.t initial electrical connections. BSFC technique applies to any string length of the PV array without additional support circuitry and results in lower voltage drops across the length of the dc cable.

An inverse problem for kernels of block Hankel operators

By Beurling–Lax–Halmos theorem the kernel of a block Hankel operator is described by an associated inner matrix. When the inner matrix is square, there is an explicit relation between the symbol function of the block Hankel operator and the inner matrix [16]. When the inner matrix is not square, little is known for the connection of the symbol function and the inner matrix. Recently, an insightful index of a matrix-valued function [18] illuminates a numerical relation between the index of the symbol and the size of the nonsquare inner matrix. In this paper, we find the explicit relation between the symbols of block Hankel operators and their associated nonsquare inner matrices. The basic result states that the symbol can be factorized into the product of a structured matrix with a certain index, otherwise free, and the adjoint of a square matrix which is a completion of the nonsquare inner matrix.

Newton’s method for the parameterized generalized eigenvalue problem with nonsquare matrix pencils

The l parameterized generalized eigenvalue problems for the nonsquare matrix pencils, proposed by Chu et al.in 2006, can be formulated as an optimization problem on a corresponding complex product Stiefel manifold. In this paper, an effective and efficient algorithm based on the Riemannian Newton’s method is established to solve the underlying problem. Under our proposed framework, to solve the corresponding Newton’s equation, it can be converted to solve a standard real symmetric linear system with a dimension reduction. By combining the Riemannian curvilinear search method with Barzilai–Borwein steps, a hybrid algorithm with both global and quadratic convergence is obtained. Numerical experiments are provided to illustrate the efficiency of the proposed method. Detailed comparisons with some latest methods are also provided to show the merits of the proposed approach.

Soft topological modes protected by symmetry in rigid mechanical metamaterials

Topological mechanics can realize soft modes in mechanical metamaterials in which the number of degrees of freedom for particle motion is finely balanced by the constraints provided by interparticle interactions. However, solid objects are generally hyperstatic (or overconstrained). Here, we show how symmetries may be applied to generate topological soft modes even in overconstrained, rigid systems. To do so, we consider non-Hermitian topology based on non-square matrices, and design a hyperstatic material in which low-energy modes protected by topology and symmetry appear at interfaces. Our approach presents a novel way of generating softness in robust scale-free architectures suitable for miniaturization to the nanoscale.

A trust‐region method for the parameterized generalized eigenvalue problem with nonsquare matrix pencils

AbstractThe l parameterized generalized eigenvalue problems for the nonsquare matrix pencils, proposed by Chu and Golub [SIAM J. Matrix Anal. Appl., 28(2006), pp. 770‐787], can be formulated as an optimization problem on a corresponding complex product Stiefel manifold. Some early proposed algorithms are based on the first‐order information of the objective function, and fast convergence could not be expected. In this article, we turn the generic Riemannian trust‐region method of Absil et al. into a practical algorithm for solving the underlying problem, which enjoys the global convergence and local superlinear convergence rate. Numerical experiments are provided to illustrate the efficiency of the proposed method. Detailed comparisons with some existing results (l = 1 and l = n) are given first. For the case l = n, the algorithm can yield exactly the same optimal solution as Ito and Murota's algorithm, which is essentially a direct method to solve the problem. The algorithm runs faster than Boutry et al.'s algorithm for the case l = 1. Further comparisons with some early proposed gradient‐based algorithms, and some latest infeasible methods for solving manifold optimization problems are also provided to show the merits of the proposed approach.

Hodge ideals for the determinant hypersurface

We determine explicitly the Hodge ideals for the determinant hypersurface as an intersection of symbolic powers of determinantal ideals. We prove our results by studying the Hodge and weight filtrations on the mixed Hodge module $$\mathcal {O}_{\mathscr {X}}(*\mathscr {Z})$$ of regular functions on the space $$\mathscr {X}$$ of $$n\times n$$ matrices, with poles along the divisor $$\mathscr {Z}$$ of singular matrices. The composition factors for the weight filtration on $$\mathcal {O}_{\mathscr {X}}(*\mathscr {Z})$$ are pure Hodge modules with underlying $$\mathcal {D}$$ -modules given by the simple $${\text {GL}}$$ -equivariant $$\mathcal {D}$$ -modules on $$\mathscr {X}$$ , where $${\text {GL}}$$ is the natural group of symmetries, acting by row and column operations on the matrix entries. By taking advantage of the $${\text {GL}}$$ -equivariance and the Cohen–Macaulay property of their associated graded, we describe explicitly the possible Hodge filtrations on a simple $${\text {GL}}$$ -equivariant $$\mathcal {D}$$ -module, which are unique up to a shift determined by the corresponding weights. For non-square matrices, $$\mathcal {O}_{\mathscr {X}}(*\mathscr {Z})$$ is replaced by the local cohomology modules $$H^{\bullet }_{\mathscr {Z}}(\mathscr {X},\mathcal {O}_{\mathscr {X}})$$ , which turn out to be pure Hodge modules. By working out explicitly the Decomposition Theorem for some natural resolutions of singularities of determinantal varieties, and using the results on square matrices, we determine the weights and the Hodge filtration for these local cohomology modules.

Projection Method for Eigenvalue Problems of Linear Nonsquare Matrix Pencils

Eigensolvers involving complex moments can determine all the eigenvalues in a given region in the complex plane and the corresponding eigenvectors of a regular linear matrix pencil. The complex moment acts as a filter for extracting eigencomponents of interest from random vectors or matrices. This study extends a projection method for regular eigenproblems to the singular nonsquare case, thus replacing the standard matrix inverse in the resolvent with the pseudoinverse. The extended method involves complex moments given by the contour integrals of generalized resolvents associated with nonsquare matrices. We establish conditions such that the method gives all finite eigenvalues in a prescribed region in the complex plane. In numerical computations, the contour integrals are approximated using numerical quadratures. The primary cost lies in the solutions of linear least squares problems that arise from quadrature points, and they can be readily parallelized in practice. Numerical experiments on large matrix pencils illustrate this method. The new method is more robust and efficient than previous methods, and based on experimental results, it is conjectured to be more efficient in parallelized settings. Notably, the proposed method does not fail in cases involving pairs of extremely close eigenvalues, and it overcomes the issue of problem size.