Matrix Pair Research Articles

Exponential stability of positive implicit dynamic equations with constant coefficients

In this paper, the problem of positivity and stability for linear time-invariant implicit dynamic equations is generally studied. We provide necessary and sufficient conditions for positivity of these equations. This characterization can be considered as a unification and generalization for some previous results. On the other hand, we study the exponential stability of positive implicit dynamic equations. Previously, this issue was not completely addressed. By using Krein–Rutman theorem, we show that a positive implicit dynamic equation on a time scale is uniformly exponentially stable if and only if the characteristic polynomial of the matrix pair defining the equation has all its coefficients of the same sign.

An iterative method for updating undamped structural systems with connectivity constraints

Model updating is a method of correcting analytical models, such as the finite element models, by improving the correlation between the measured data and the analytical modal data. In this paper, we develop an iterative method to update undamped structural systems by using measured modal parameters. The method allows mass and stiffness matrices to be updated simultaneously. And we can obtain optimal updated matrices within finite iteration steps by choosing a special kind of initial matrix pair. More importantly, the method can preserve the physical connectivity of the original model and the updated model is in good agreement with the experimental modal data. Two real numerical examples show that the proposed method is feasible and efficient.

Refined and refined harmonic Jacobi–Davidson methods for computing several GSVD components of a large regular matrix pair

Refined and refined harmonic Jacobi–Davidson methods for computing several GSVD components of a large regular matrix pair

SFINN: inferring gene regulatory network from single-cell and spatial transcriptomic data with shared factor neighborhood and integrated neural network.

The rise of single-cell RNA sequencing (scRNA-seq) technology presents new opportunities for constructing detailed cell type-specific gene regulatory networks (GRNs) to study cell heterogeneity. However, challenges caused by noises, technical errors, and dropout phenomena in scRNA-seq data pose significant obstacles to GRN inference, making the design of accurate GRN inference algorithms still essential. The recent growth of both single-cell and spatial transcriptomic sequencing data enables the development of supervised deep learning methods to infer GRNs on these diverse single-cell datasets. In this study, we introduce a novel deep learning framework based on shared factor neighborhood and integrated neural network (SFINN) for inferring potential interactions and causalities between transcription factors and target genes from single-cell and spatial transcriptomic data. SFINN utilizes shared factor neighborhood to construct cellular neighborhood network based on gene expression data and additionally integrates cellular network generated from spatial location information. Subsequently, the cell adjacency matrix and gene pair expression are fed into an integrated neural network framework consisting of a graph convolutional neural network and a fully-connected neural network to determine whether the genes interact. Performance evaluation in the tasks of gene interaction and causality prediction against the existing GRN reconstruction algorithms demonstrates the usability and competitiveness of SFINN across different kinds of data. SFINN can be applied to infer GRNs from conventional single-cell sequencing data and spatial transcriptomic data. SFINN can be accessed at GitHub: https://github.com/JGuan-lab/SFINN.

Quantum Size-Driven Spectral Variations in Pillar[n]arene Systems: A Density Functional Theory and Wave Function Assessment.

This study explores the quantum size effects on the optical properties of pillar[n]arene (n = 5, 6, 7, 8) utilizing density functional theory (DFT) and wave function analysis. The mechanisms of electron transitions in one-photon absorption (OPA) and two-photon absorption (TPA) spectra are investigated, alongside the calculation of electron circular dichroism (ECD) for these systems. Transition Density Matrix (TDM) and electron-hole pair density maps are employed to study the electron excitation characteristics, unveiling a notable size dependency. Analysis of the transition electric dipole moment (TEDM) and the transition magnetic dipole moment (TMDM) reveals the electromagnetic interaction mechanism within pillar[n]arene. Raman spectra computations further elucidate vibrational modes, while interactions with external environments are studied using electrostatic potential (ESP) analysis, and electron delocalization is assessed under an external magnetic field, providing insights into the magnetically induced current phenomena within these supramolecular structures. The thermal stability of pillar[n]arene was investigated by ab initio molecular dynamics (AIMD).

Eigenvalue characterization of some structured matrix pencils under linear perturbation

We study the effect of linear perturbations on three families of matrix pencils. The matrix pairs of the first two families are Hermitian/skew-Hermitian with special $3\times 3$ block cases appeared in continuous-time control, and the matrix pairs of the third family are special $3\times 3$ non-Hermitian block matrices appeared in discrete-time control. For the first family of matrix pencils and more general cases of the second family of matrix pencils, based on the properties of the involved matrices, we obtain some upper or lower bounds on the set of eigenvalues of linearly perturbed matrix pencils which are on the imaginary axis. Studying a special $3\times 3$ block matrix pencil, which is associated with continuous-time control, leads us to some linear perturbation that do not preserve (properly) the structure of the matrices. This, in turn, leads to a numerical technique for finding the nearest Hermitian/skew-Hermitian matrix pencil which can satisfy conditions such that, for some nonzero real perturbation parameter, some or all of its eigenvalues lie on the imaginary axis. We also study the linearly perturbed matrix pencils, associated with discrete-time control, using an one-to-one equivalence between the matrix pencil of continuous-time problem and the matrix pencil of discrete-time problem.

Double-variable trace maximization for extreme generalized singular quartets of a matrix pair: A geometric method

In this paper, we consider the problem of computing an arbitrary generalized singular value of a Grassman or real matrix pair and a triplet of associated generalized singular vectors. Based on the QR factorization, the problem is reformulated as two novel trace maximization problems, each of which has double variables with unitary constraints or orthogonal constraints. Theoretically, we show that the arbitrarily prescribed extreme generalized singular values and associated triplets of generalized singular vectors can be determined by the global solutions of the constrained trace optimization problems. Then we propose a geometric inexact Newton–conjugate gradient (Newton-CG) method for solving their equivalent trace minimization problems over the Riemannian manifold of all fixed-rank partial isometries. The proposed method can extract not only the prescribed extreme generalized singular values but also associated triplets of generalized singular vectors. Under some mild assumptions, we establish the global and quadratic convergence of the proposed method. Finally, numerical experiments on both synthetic and real data sets show the effectiveness and high accuracy of our method.

Scale perception data compression method with blockchain technology for UAV communication

The demand for high data throughput and security in Unmanned Aerial Vehicle (UAV) communication is rising. Efficient data transmission remains challenging due to limited wireless resources. This paper introduces a Scale Perception Compression Method that leverages compression and blockchain technology to optimize UAV communication. The method involves normalizing data into matrices, generating compression and decompression matrix codebook pairs, and selecting suitable pairs for small or large-scale matrices. Simulation results show that the error rate is associated with the distributions of the data and compression matrix. It remains relatively low when the data and compression matrix subjected to Uniform distribution compared to Gaussian distribution, regardless of the values of the number of rows m and columns r in compression matrix. Bubble method is more suitable to find out the proper matrix pair to compress and decompress transmitted data for small-scale codebook because it is stable, fast and results in lower difference between original and recovered data matrices. The same applies to Merge method for the large-scale codebook. The ER is always lower based on a small-scale codebook searching by Bubble method than that of on a large-scale codebook searching by Merge method, which indicates the former is sufficient for compression.

The Joint Bidiagonalization of a Matrix Pair with Inaccurate Inner Iterations

The Joint Bidiagonalization of a Matrix Pair with Inaccurate Inner Iterations

The Hermitian solution to a matrix inequality under linear constraint

<abstract><p>In this paper, the necessary and sufficient conditions under which the matrix inequality $ C^*XC\geq D\ (>D) $ subject to the linear constraint $ A^*XA = B $ is solvable are deduced by means of the spectral decompositions of some matrices and the generalized singular value decomposition of a matrix pair. An explicit expression of the general Hermitian solution is also provided. One numerical example demonstrates the effectiveness of the proposed method.</p></abstract>

On a Particular Solution of the σ-Commutation Problem () for Toeplitz and Hankel Matrices

A unified approach is proposed to the construction of matrix pairs (T,H) that solve the ‑commutation problem for Toeplitz and Hankel matrices. For a certain particular case, a family of solutions is derived.

A coupled piezo-triboelectric nanogenerator based on the electrification of biaxially oriented polyethylene terephthalate food packaging films

This study reports that biaxially oriented silica-filled polyethylene terephthalate (BOPET) films used worldwide as food packaging materials combine unique dual electrification properties that enable the effective conversion of mechanical energy into electricity. The electricity output depends on the physical contact and separation of the polymeric matrix and triboelectric pair material with different affinities for electrons on its surface. To exploit the dual piezoelectric and triboelectric electrification of the films materials, a hybrid piezoelectric triboelectric nanogenerator energy harvester using this film as an electron source was assembled and tested. When BOPET film is paired with polyvinylidene fluoride (PVDF) nanofibre network, which due to the electronegativity difference gained electrons, the performance of the nanogenerator to the output open-circuit increases voltage values above 305 V, and the short-circuit current density and charge density to 12 ± 1μAcm−2 and 37 ± 2μCm−2, respectively. Further, underlaying the PVDF tribo-contact layer by a dielectric low permittivity spunbond non-woven polypropylene raises the performance of the nanogenerator more than doubled (624 V) with corresponding densities of 61 ± 7μAcm−2 and 101 ± 4μCm−2.

Thick-restarted joint Lanczos bidiagonalization for the GSVD

The computation of the partial generalized singular value decomposition (GSVD) of large-scale matrix pairs can be approached by means of iterative methods based on expanding subspaces, particularly Krylov subspaces. We consider the joint Lanczos bidiagonalization method, and analyze the feasibility of adapting the thick restart technique that is being used successfully in the context of other linear algebra problems. Numerical experiments illustrate the effectiveness of the proposed method. We also compare the new method with an alternative solution via equivalent eigenvalue problems, considering accuracy as well as computational performance. The analysis is done using a parallel implementation in the SLEPc library.

Qualitative sign stability of linear time invariant descriptor systems

This article discusses assessing the instability of a continuous linear homogeneous timeinvariant descriptor system. Some necessary conditions and sufficient conditions are derived to establish the stability of a matrix pair by the fundamentals of qualitative ecological principles. The proposed conditions are derived using only the qualitative (sign) information of the matrix pair elements. Based on these conditions, the instability of a matrix pair can easily be determined, without any magnitude information of the matrix pair elements and without numerical eigenvalues calculations. With the proposed theory, Magnitude Dependent Stable, Magnitude Dependent Unstable, and Qualitative Sign Stable matrix pairs can be distinguished. The consequences of the proposed conditions and some illustrative examples are discussed.

Properties and characterizations of dual sharp orders

In this paper, we extend the sharp order to dual ring. To be specific, we introduce D-sharp, C-sharp, G- and G-sharp binary relations of dual matrices, and discuss relevant properties of these dual binary relations by using dual group generalized inverse(DGGI for short), group dual generalized inverse(GDGI for short) and matrix decomposition. We conclude that D-sharp and G-sharp binary relations are partial orders. The dual binary relations are connected with the sharp order by using the upper triangular block matrix. Furthermore, we discuss relationships among dual binary relations mentioned above. When dual parts of dual matrix pairs are all zero matrices, dual binary relations(D-sharp, C-sharp, G- and G-sharp binary relations) are equivalent to the sharp order.

Direct and Inverse Problems of String Equation by Numerov’s Method

It is well known that free vibration of a taut string having mass per unit m(x) and frequency $$\omega$$ is governed by ordinary differential equation $$y^{\prime \prime }+\omega ^2m(x)y=0.$$ In this paper, first we discretize the differential equation by using Numerov’s method to obtain a matrix eigenvalue problem of the form $$-Au=\Lambda B M u$$ , where A and B are constant tridiagonal matrices and M is a diagonal matrix related to mass function m(x). In direct problem, for a given m(x), we approximate the first N eigenvalues of the string equation by making a new correction on the eigenvalues of matrix pair $$(-A,BM)$$ . Also we obtain the error order of corrected eigenvalues. For inverse problem, we propose an efficient algorithm for constructing unknown mass function m(x) by using given spectra by solving a nonlinear system. We solve the nonlinear system by using modified Newton’s method and a regularization technique. The convergence of Newton’s method is proved. Finally, we give some numerical examples to illustrate the efficiency of the proposed algorithm.

Length Function and Simultaneous Triangularization of Matrix Pairs

The paper interrelates the simultaneous triangularization problem for matrix pairs with the Paz problem and known results on the length of the matrix algebra. The length function is applied to the Al’pin–Koreshkov algorithm, and it is demonstrated how its multiplicative complexity can be reduced. An asymptotically superior procedure for verifying the simultaneous triangularizability of a pair of complex matrices is provided. The procedure is based on results on the lengths of upper triangular matrix algebras. Also the definition of the hereditary length of an algebra is introduced, and the problem of computing the hereditary lengths of matrix algebras is discussed.

On the invertibility indices and the Morse normal form for linear multivariable systems

In this paper, by extending the controllability and Brunovsky indices for a matrix pair to a matrix quadruple, three kinds of indices, say, the invertibility indices, the left invertibility indices, and the right invertibility indices, for linear time-invariant systems are introduced and studied. Based on properties of these indices, a neat expression for the Morse lists of a linear system is given without transforming it into its Morse normal form, which is a deep and fundamental result in linear systems theory, and takes an important role in many analysis and design problems for linear systems. Furthermore, the Morse normal form with only algebraic equivalence transformation is also investigated and is simplified.

The Joint Bidiagonalization Method for Large GSVD Computations in Finite Precision

The joint bidiagonalization (JBD) method has been used to compute some extreme generalized singular values and vectors of a large regular matrix pair . We make a numerical analysis of the underlying JBD process and establish relationships between it and two mathematically equivalent Lanczos bidiagonalizations in finite precision. Based on the results of numerical analysis, we investigate the convergence of the approximate generalized singular values and vectors of . The results show that, under some mild conditions, the semiorthogonality of Lanczos-type vectors suffices to deliver approximate generalized singular values with the same accuracy as the full orthogonality does, meaning that it is only necessary to seek for efficient semiorthogonalization strategies for the JBD process. We establish a sharp bound for the residual norm of an approximate generalized singular value and corresponding approximate right generalized singular vectors, which can reliably estimate the residual norm without explicitly computing the approximate right generalized singular vectors before the convergence occurs.

Optimal control and stabilization for linear mean‐field system with indefinite quadratic cost functional

AbstractIn this paper, we consider an optimal control and stabilization problem of linear mean‐field (MF) system, where the quadratic cost functional is allowed to be indefinite. Inspired by the equivalent cost functional method, we introduce a subset, which helps us to investigate the convergence property of generalized differential Riccati equations arising in indefinite mean‐field linear quadratic (MF‐LQ) problems with finite horizon. A coupled generalized algebraic Riccati equation (GARE) is thus obtained. More importantly, the solution pair of corresponding GARE can be decomposed into a solution pair of a coupled standard algebraic Riccati equation (SARE) and a matrix pair in the subset we introduce. Thus, an equivalent relationship is established between GARE and SARE. With this equivalence, we derive necessary and sufficient conditions to stabilize linear MF‐system with indefinite weighting matrices in mean‐square sense.