Diagonal Matrix Research Articles

Measuring, processing, and generating partially coherent light with self-configuring optics

Optical phenomena always display some degree of partial coherence between their respective degrees of freedom. Partial coherence is of particular interest in multimodal systems, where classical and quantum correlations between spatial, polarization, and spectral degrees of freedom can lead to fascinating phenomena (e.g., entanglement) and be leveraged for advanced imaging and sensing modalities (e.g., in hyperspectral, polarization, and ghost imaging). Here, we present a universal method to analyze, process, and generate spatially partially coherent light in multimode systems by using self-configuring optical networks. Our method relies on cascaded self-configuring layers whose average power outputs are sequentially optimized. Once optimized, the network separates the input light into its mutually incoherent components, which is formally equivalent to a diagonalization of the input density matrix. We illustrate our method with numerical simulations of Mach-Zehnder interferometer arrays and show how this method can be used to perform partially coherent environmental light sensing, generation of multimode partially coherent light with arbitrary coherency matrices, and unscrambling of quantum optical mixtures. We provide guidelines for the experimental realization of this method, including the influence of losses, paving the way for self-configuring photonic devices that can automatically learn optimal modal representations of partially coherent light fields.

Statistics of Matrix Elements of Local Operators in Integrable Models

We study the statistics of matrix elements of local operators in the basis of energy eigenstates in a paradigmatic, integrable, many-particle quantum theory, the Lieb-Liniger model of bosons with repulsive delta-function interactions. Using methods of quantum integrability, we determine the scaling of matrix elements with system size. As a consequence of the extensive number of conservation laws, the structure of matrix elements is fundamentally different from, and much more intricate than, the predictions of the eigenstate thermalization hypothesis for generic models. We uncover an interesting connection between this structure for local operators in interacting integrable models and the one for local operators that are not local with respect to the elementary excitations in free theories. We find that typical off-diagonal matrix elements ⟨μ|O|λ⟩ in the same macrostate scale as exp(−cOLln(L)−LMμ,λO), where the probability distribution function for Mμ,λO is well described by Fréchet distributions and cO depends only on macrostate information. In contrast, typical off-diagonal matrix elements between two different macrostates scale as exp(−dOL2), where dO depends only on macrostate information. Diagonal matrix elements depend only on macrostate information up to finite-size corrections. Published by the American Physical Society 2024

On frame diagonalization of square matrices

In this paper, we introduce a frame diagonalization of matrices in M n ( C ) , called QR-frame diagonalization, by using QR-factorization. Furthermore, we show that every matrix A in M n ( C ) is QR-frame diagonalizable. Also we introduce another frame diagonalization via Hadamard matrices in M n ( R ) , called Hadamard frame diagonalization, by using Hadamard matrices for n = 2, 4 or multiple of 4. We show that every matrix A in M n ( R ) which n = 2, 4 or multiple of 4 is Hadamard frame diagonalizable. In this frame diagonalization, we can find entries on main diagonal Δ by using inner product. Moreover we introduce frame diagonalization of matrices on left quaternionic Hilbert spaces M n ( H ) .

Piecewise modified Prandtl-Ishlinskii model for precise nano-positioning

Piezoelectric ceramic, an indispensable micro-nano actuator, has advantages of small size, high sensitivity and strong controllability, which is widely applied in various fields like micro-nano processing, medicine, chemistry and materials. However, its nonlinear and asymmetric hysteresis features directly affect its positioning accuracy. The classical and generalized Prandtl-Ishlinskii models with linear symmetry operators are commonly used to deal with these issues, but they still face difficulties in asymmetric systems. Therefore, a piecewise modified Prandtl-Ishlinskii (PMPI) model is proposed to compensate nonlinear and asymmetric hysteresis in this paper. The new nonlinear operator and piecewise nonlinear envelope function are introduced into the PMPI model. Independent right and left parts are adopted to describe hysteresis increasing and decreasing stages, while the threshold and weight vectors correspond to them. Weight vectors are extended and a new diagonal matrix is constructed for avoiding excessive parameters and computation. Simulations and experiments are shown that the PMPI model has good compensation performance and effectiveness, which has better fitting, higher accuracy, fewer parameters and computation, and can be well applied to the precise nano-positioning systems with less nonlinear and asymmetric errors.

The Distributed Adaptive Bipartite Consensus Tracking Control of Networked Euler–Lagrange Systems with an Application to Quadrotor Drone Groups

Actuator faults and external disturbances, which are inevitable due to material fatigue, operational wear and tear, and unforeseen environmental impacts, cause significant threats to the control reliability and performance of networked systems. Therefore, this paper primarily focuses on the distributed adaptive bipartite consensus tracking control problem of networked Euler–Lagrange systems (ELSs) subject to actuator faults and external disturbances. A robust distributed control scheme is developed by combining the adaptive distributed observer and neural-network-based tracking controller. On the one hand, a new positive definite diagonal matrix associated with an asymmetric Laplacian matrix is constructed in the distributed observer, which can be used to estimate the leader’s information. On the other hand, neural networks are adopted to approximate the lumped uncertainties composed of unknown matrices and external disturbances in the follower model. The adaptive update laws are designed for the unknown parameters in neural networks and the actuator fault factors to ensure the boundedness of estimation errors. Finally, the proposed control scheme’s effectiveness is validated through numerical simulations using two types of typical ELS models: two-link robot manipulators and quadrotor drones. The simulation results demonstrate the robustness and reliability of the proposed control approach in the presence of actuator faults and external disturbances.

Improved dynamic performance of triple active bridge DC-DC converter using differential flatness control for more electric aircraft applications

The triple active bridge (TAB) converter has been proposed to improve power density and availability in more electric aircraft applications. However, the conventional proportional-integral (PI) controller used for TAB voltage control often exhibits slow response and significant overshoot, which can impact dynamic performance. Moreover, achieving decoupling control among ports introduces computational complexity in controller design. To address these issues, this paper introduces an effective diagonal matrix decoupling strategy based on differential flatness control with single-phase shift modulation applied to a TAB converter. The proposed differential flatness control offers superior transient performance, control flexibility, and precision, ensuring compliance with DC voltage regulations while achieving optimal decoupling among ports. The comparative analysis assesses various criteria, including steady-state and dynamic performance, control complexity, and regulation and tracking properties, against PI control, unit matrix decoupling control, and the proposed differential flatness control. Hardware-in-loop (HIL) experimentation verifies the performance, confirming faster dynamic characteristics and robust port power decoupling. The results show a significant improvement in efficiency, reaching a maximum of 95.5 %, indicating reduced switching losses and enhanced overall system performance. The study concludes with a discussion on the implications of these findings and potential future research directions, such as integrating advanced optimization algorithms further to enhance the control strategy's robustness and adaptability.

On Quark–Lepton Mixing and the Leptonic CP Violation

In the absence of a Grand Unified Theory framework, connecting the values of the mixing parameters in the quark-and-lepton sector is a difficult task, unless one introduces ad hoc relations among the matrices that diagonalize such different kinds of fermions. In this paper, we discuss in detail the possibility that the PMNS matrix is given by the product UPMNS=VCKM★T★, where T comes from the diagonalization of a see-saw like mass matrix that can be of a Bimaximal (BM), Tri-Bimaximal (TBM) and Golden Ratio (GR) form, and identify the leading corrections to such patterns that allow for a good fit to the leptonic mixing matrix as well as to the CP phase. We also show that the modified versions of BM, TBM and GR can easily accommodate the solar and atmospheric mass differences.

Robust Direction-of-Arrival Estimation in the Presence of Outliers and Noise Nonuniformity

In direction-of-arrival (DOA) estimation with sensor arrays, the background noise is usually modeled to be uncorrelated uniform white noise, such that the related algorithms can be greatly simplified by making use of the property of the noise covariance matrix being a diagonal matrix with identical diagonal entries. However, this model can be easily violated by the nonuniformity of sensor noise and the presence of outliers that may arise from unexpected impulsive noise. To tackle this problem, we first introduce an exploratory factor analysis (EFA) model for DOA estimation in nonuniform noise. Then, to deal with the outliers, a generalized extreme Studentized deviate (ESD) test is applied for outlier detection and trimming. Based on the trimmed data matrix, a modified EFA model, which belongs to weighted least-squares (WLS) fitting problems, is presented. Furthermore, a monotonic convergent iterative reweighted least-squares (IRLS) algorithm, called the iterative majorization approach, is introduced to solve the WLS problem. Simulation results show that the proposed algorithm offers improved robustness against nonuniform noise and observation outliers over traditional algorithms.

Dynamically accelerating the power iteration with momentum

AbstractIn this article, we propose, analyze and demonstrate a dynamic momentum method to accelerate power and inverse power iterations with minimal computational overhead. The method can be applied to real diagonalizable matrices, is provably convergent with acceleration in the symmetric case, and does not require a priori spectral knowledge. We review and extend background results on previously developed static momentum accelerations for the power iteration through the connection between the momentum accelerated iteration and the standard power iteration applied to an augmented matrix. We show that the augmented matrix is defective for the optimal parameter choice. We then present our dynamic method which updates the momentum parameter at each iteration based on the Rayleigh quotient and two previous residuals. We present convergence and stability theory for the method by considering a power‐like method consisting of multiplying an initial vector by a sequence of augmented matrices. We demonstrate the developed method on a number of benchmark problems, and see that it outperforms both the power iteration and often the static momentum acceleration with optimal parameter choice. Finally, we present and demonstrate an explicit extension of the algorithm to inverse power iterations.

Fitting of Coupled Potential Energy Surfaces via Discovery of Companion Matrices by Machine Intelligence.

Fitting coupled potential energy surfaces is a critical step in simulating electronically nonadiabatic chemical reactions and energy transfer processes. Analytic representation of coupled potential energy surfaces enables one to perform detailed dynamics calculations. Traditionally, fitting is performed in a diabatic representation to avoid fitting the cuspidal ridges of coupled adiabatic potential energy surfaces at conical intersection seams. In this work, we provide an alternative approach by carrying out fitting in the adiabatic representation using a modified version of the Frobenius companion matrices, whose usage was first proposed by Opalka and Domcke. Their work involved minimizing the errors in fits of the characteristic polynomial coefficients (CPCs) and diagonalizing the resulting companion matrix, whose eigenvalues are adiabatic potential energies. We show, however, that this may lead to complex eigenvalues and spurious discontinuities. To alleviate this problem, we provide a new procedure for the automatic discovery of CPCs and the diagonalization of a companion matrix by using a special neural network architecture. The method effectively allows analytic representation of global coupled adiabatic potential energy surfaces and their gradients with only adiabatic energy input and without experience-based selection of a diabatization scheme. We demonstrate that the new procedure, called the companion matrix neural network (CMNN), is successful by showing applications to LiH, H3, phenol, and thiophenol.

An Inverse Matrix Eigenvalue Problem for Constructing a Vibrating Rod

Abstract The free longitudinal vibrations of a rod are described by a differential equation of the form ( P ( x ) y ′ ) ′ + λ P ( x ) y ( x ) = 0 {(P(x)y\prime)^{\prime}+\lambda P(x)y(x)=0} , where P ⁢ ( x ) {P(x)} is the cross section area at point x and λ is an eigenvalue parameter. In this paper, first we discretize this differential equation by using the finite difference method to obtain a matrix eigenvalue problem of the form 𝐀 ⁢ Y = Λ ⁢ 𝐁 ⁢ Y {\mathbf{A}Y=\Lambda\mathbf{B}Y} , where 𝐀 {\mathbf{A}} and 𝐁 {\mathbf{B}} are Jacobi and diagonal matrices dependent to cross section P ⁢ ( x ) {P(x)} , respectively. Then we estimate the eigenvalues of the rod equation by correcting the eigenvalues of the resulting matrix eigenvalue problem. We give a method based on a correction idea to construct the cross section P ⁢ ( x ) {P(x)} by solving an inverse matrix eigenvalue problem. We give some numerical examples to illustrate the efficiency of the proposed method. The results show that the convergence order of the method is O ⁢ ( h 2 ) {O(h^{2})} .

Unconditionally convergent [formula omitted] splitting iterative methods for variable coefficient Riesz space fractional diffusion equations

In this paper, we consider fast solvers for discrete linear systems generated by Riesz space fractional diffusion equations. We extract a scalar matrix, a compensation matrix, and a τ matrix from the coefficient matrix, and use their sum to construct a class of τ splitting iterative methods. Additionally, we design a preconditioner for the conjugate gradient method. Theoretical analyses show that the proposed τ splitting iterative methods are unconditionally convergent with convergence rates independent of step-sizes. Numerical results are provided to demonstrate the effectiveness of the proposed iterative methods.

Fourier Transforms in Digital Image Processing Courses

To help teaching and learning of Fourier transforms in digital image processing courses, an approach to the Fourier transforms from a standpoint of linear algebra is presented. After representing a periodic sequence by a circulant matrix, the periodic convolution can be formulated in terms of matrix multiplication, and finally, the Fourier transform is written in a form of matrix diagonalization. As a comparison, summation formulas are prevalent in traditional courses on signals and systems and matrices are scarcely found in textbooks of digital signal processing. We hope that this approach will make the Fourier transform easier to teach and learn in digital image processing courses.

Non-commutative probability insights into the double-scaling limit SYK model with constant perturbations: moments, cumulants and q-independence

Extending the results of Wu (2022 J. Phys. A 55 415207), we study the double-scaling limit Sachdev–Ye–Kitaev model with an additional diagonal matrix with a fixed number c of nonzero constant entries θ. This constant diagonal term can be rewritten in terms of Majorana fermion products. Its specific formula depends on the value of c. We find exact expressions for the moments of this model. More importantly, by proposing a moment-cumulant relation, we reinterpret the effect of introducing a constant term in the context of non-commutative probability theory. This gives rise to a q~ dependent mixture of independences within the moment formula. The parameter q~ , derived from the q-Ornstein–Uhlenbeck (q-OU) process, controls this transformation. It interpolates between classical independence ( q~=1 ) and Boolean independence ( q~=0 ). The underlying combinatorial structures of this model provide the non-commutative probability connections. Additionally, we explore the potential relation between these connections and their gravitational path integral counterparts.

On the Signless Laplacian ABC-Spectral Properties of a Graph

In the paper, we introduce the signless Laplacian ABC-matrix Q̃(G)=D¯(G)+Ã(G), where D¯(G) is the diagonal matrix of ABC-degrees and Ã(G) is the ABC-matrix of G. The eigenvalues of the matrix Q̃(G) are the signless Laplacian ABC-eigenvalues of G. We give some basic properties of the matrix Q̃(G), which includes relating independence number and clique number with signless Laplacian ABC-eigenvalues. For bipartite graphs, we show that the signless Laplacian ABC-spectrum and the Laplacian ABC-spectrum are the same. We characterize the graphs with exactly two distinct signless Laplacian ABC-eigenvalues. Also, we consider the problem of the characterization of the graphs with exactly three distinct signless Laplacian ABC-eigenvalues and solve it for bipartite graphs and, in some cases, for non-bipartite graphs. We also introduce the concept of the trace norm of the matrix Q̃(G)−tr(Q̃(G))nI, called the signless Laplacian ABC-energy of G. We obtain some upper and lower bounds for signless Laplacian ABC-energy and characterize the extremal graphs attaining it. Further, for graphs of order at most 6, we compare the signless Laplacian energy and the ABC-energy with the signless Laplacian ABC-energy and found that the latter behaves well, as there is a single pair of graphs with the same signless Laplacian ABC-energy unlike the 26 pairs of graphs with same signless Laplacian energy and eight pairs of graphs with the same ABC-energy.

A novel weighted sparse classification framework with extended discriminative dictionary for data-driven bearing fault diagnosis

Dictionary learning has emerged as an effective approach for data-driven fault diagnosis due to its strong sparse representation ability. Nevertheless, the gathered vibration signals always exhibit a time-shift phenomenon, greatly diminishing the representation capability of dictionary learning methods. Besides, in real-world industrial scenarios, where heavy noise as common features inevitably cover the discriminative features, the decline in the recognition performance of data-driven fault diagnosis methods is a critical challenge. In this paper, a novel weighted sparse classification framework with extended discriminative dictionary (WSC-EDD) is proposed. First, an extended discriminative dictionary design strategy is developed to construct generalized-domain extended dictionary model with strong representation and discrimination ability, in which a novel generalized-domain dictionary fusion method is developed to reduce the effect of time-shift phenomenon and enhance the representation ability of the dictionary. Further, generalized-domain discriminative sub-dictionaries are optimized by the K-singular value decomposition in a data-driven fashion and then the whole extension dictionary are designed adaptively. Second, a weighted optimization method is developed to highlight the contribution of non-zero elements in the correct projection region in sparse codes, in which weighted diagonal matrices are designed. Finally, a sparse recognition method is established for bearing fault classification. The experiment results on three challenging bearing datasets demonstrate that the proposed framework yields average diagnosis accuracies of 99.75%, 99.93% and 100%, respectively. Moreover, the superiority and robustness of WSC-EDD are further confirmed by comparing with several competing methods.

Experimental demonstration of magnetic tunnel junction-based computational random-access memory

The conventional computing paradigm struggles to fulfill the rapidly growing demands from emerging applications, especially those for machine intelligence because much of the power and energy is consumed by constant data transfers between logic and memory modules. A new paradigm, called “computational random-access memory (CRAM),” has emerged to address this fundamental limitation. CRAM performs logic operations directly using the memory cells themselves, without having the data ever leave the memory. The energy and performance benefits of CRAM for both conventional and emerging applications have been well established by prior numerical studies. However, there is a lack of experimental demonstration and study of CRAM to evaluate its computational accuracy, which is a realistic and application-critical metric for its technological feasibility and competitiveness. In this work, a CRAM array based on magnetic tunnel junctions (MTJs) is experimentally demonstrated. First, basic memory operations, as well as 2-, 3-, and 5-input logic operations, are studied. Then, a 1-bit full adder with two different designs is demonstrated. Based on the experimental results, a suite of models has been developed to characterize the accuracy of CRAM computation. Scalar addition, multiplication, and matrix multiplication, which are essential building blocks for many conventional and machine intelligence applications, are evaluated and show promising accuracy performance. With the confirmation of MTJ-based CRAM’s accuracy, there is a strong case that this technology will have a significant impact on power- and energy-demanding applications of machine intelligence.

Applying matrix diagonalisation in the classroom with GeoGebra: parametrising the intersection of a sphere and plane

In this paper, we explore how the typical second year undergraduate topic of matrix diagonalisation can be applied to solve a geometric problem which arises frequently in Vector Calculus: to find the curve of intersection of a plane and a surface. We use GeoGebra, which offers a cost-free and dynamic interface, to explore a specific variant of this problem, namely the parametrisation of the intersection of a plane and sphere. GeoGebra not only allows students to confirm their calculations, but it can also provide motivation to create connections between prior learning experiences and new knowledge.

Lattice realizations of topological defects in the critical (1+1)-d three-state Potts model

Topological/perfectly-transmissive defects play a fundamental role in the analysis of the symmetries of two dimensional conformal field theories (CFTs). In the present work, spin chain regularizations for these defects are proposed and analyzed in the case of the three-state Potts CFT. In particular, lattice versions for all the primitive defects are presented, with the remaining defects obtained from the fusion of the primitive ones. The defects are obtained by introducing modified interactions around two given sites of an otherwise homogeneous spin chain with periodic boundary condition. The various primitive defects are topological on the lattice except for one, which is topological only in the scaling limit. The lattice models are analyzed using a combination of exact diagonalization and density matrix renormalization group techniques. Low-lying energy spectra for different defect Hamiltonians as well as entanglement entropy of blocks located symmetrically around the defects are computed. The latter provides a convenient way to compute the g-function which characterizes various defects. Finally, the eigenvalues of the line operators in the “crossed channel” and fusion of different defect lines are also analyzed. The results are all in agreement with expectations from conformal field theory.

Radiative lifetime of the A Π1/22 state in RaF with relevance to laser cooling

The radiative lifetime of the AΠ1/22 (v=0) state in radium monofluoride (RaF) is measured to be 35(1) ns. The lifetime of this state and the related decay rate Γ=2.86(8)×107 s−1 are of relevance to the laser cooling of RaF via the optically closed AΠ1/22←XΣ1/22 transition, which makes the molecule a promising probe to search for new physics. RaF is found to have a comparable photon-scattering rate to homoelectronic laser-coolable molecules. Owing to its highly diagonal Franck-Condon matrix, it is expected to scatter an order of magnitude more photons than other molecules when using just three cooling lasers, before it decays to a dark state. The lifetime measurement in RaF is benchmarked by measuring the lifetime of the 8P3/2 state in Fr to be 83(3) ns, in agreement with literature. Published by the American Physical Society 2024