Symmetric Matrix Research Articles

ABSTRACT We develop new, easily computable exponential decay bounds for inverses of banded matrices, based on the quasiseparable representation of Green matrices. The bounds rely on a diagonal dominance hypothesis and do not require explicit spectral information. Numerical experiments and comparisons show that these new bounds can be advantageous, especially for nonsymmetric or symmetric indefinite matrices.

The real symmetric matrices with a given rank and a P-set with maximum size

Manifold Learning Based on Locally Linear Embedding for Symmetric Positive Definite Matrix

The nearest convex hull (NCH) classifier is a promising algorithm for the classification of biosignals, such as electroencephalography (EEG) signals, especially when adapted to the classification of symmetric positive definite matrices. In this paper, we implemented a version of this classifier that can execute either on a traditional computer or a quantum simulator, and we tested it against state-of-the-art classifiers for EEG classification. This article addresses the practical challenges of adapting a classical algorithm to one that can be executed on a quantum computer or a quantum simulator. One of these challenges is to find a formulation of the classification problem that is quadratic, is binary, and accepts only linear constraints—that is, an objective function that can be solved using a variational quantum algorithm. In this article, we present two approaches to solve this problem, both compatible with continuous variables. Finally, we evaluated, for the first time, the performance of the NCH classifier on real EEG data using both quantum and classical optimization methods. We selected a particularly challenging dataset, where classical optimization typically performs poorly, and demonstrated that the nearest convex hull classifier was able to generalize with a modest performance. One lesson from this case study is that, by separating the objective function from the solver, it becomes possible to allow an existing classical algorithm to run on a quantum computer, as long as an appropriate objective function—quadratic and binary—can be found.

Abstract For vivid, immersive overlay of virtual images onto background scenes in augmented reality (AR) applications, it is crucial for the display element to achieve controllability of spectral selectivity and transmittance level. At the current stage, the transmittance of self-emissive transparent displays is limited to at most ∼60 %, constrained by the fill factor of emissive regions, restricting their scalability for immersive experiences. Although projection-based transparent screens using frequency-selective scatterers offer a promising alternative, the platforms suffer from spectral broadening and instability originating from color-dependent scattering and inter-scatterer coupling. Here, we present a transparent screen architecture based on multicolor nanoring arrays. By tuning the nanoring’s resonance via inner-aperture size engineering, the architecture enables dense, symmetric RGB arrays with isolated and homogenized scattering responses. For inter-scatterer distances of 100–200+ nm, full-wave simulations confirm the robustness of well-isolated RGB reflections (FWHM < 25 nm), along with exceptional tunability of transmittance (50 % to above 80 %). As a platform for AR displays, we demonstrate the widest reported transparency-control range without any penalty to color balance or spectral selectivity. We also analyze the gamut area of projected images across transmittance levels, achieving a net gamut expansion (+11.0 % p at Λ = 120 nm; +5.5 % p at Λ = 190 nm) from the spectral narrowing of projection sources, and further propose a practical design map linking the maximum allowable transmittance to the ambient-to-source noise ratio. Our nanoring-based architecture provides a robust and scalable platform for next-generation transparent displays under real-world lighting conditions.

Transcranial magnetic stimulation (TMS) is a rapidly advancing neuromodulation technology whose efficacy depends directly on the spatial distribution of induced electric fields (E-fields). Traditional planar TMS coils with large diameters can increase stimulation intensity but at the cost of reduced focalization. To address this trade-off, we propose an eccentric curved symmetric array (ECS array), which integrates inner planar coils with smaller radii and outer curved coils with larger radii. Together, they form an eccentric symmetric structure. Depending on the bending direction of the outer coils, the ECS array can be classified as upward bending (U-ECS) or downward bending. Finite element analysis of the ECS array demonstrates a bounded, highly focused E-field distribution on the target plane 2 cm below the coil. Compared with the conventional figure-of-eight coil, the optimized U-ECS array enhances stimulation intensity by 63.29% and reduces the focusing area by 12.33%. To validate these findings, we built an ECS prototype with a dedicated TMS stimulation circuit and measured the induced E-fields. The experimental results closely matched simulations, confirming that the ECS array can deliver stronger and more focused stimulation fields. This design provides a promising strategy for non-invasive neural modulation, with potential applications in therapeutic interventions and cognitive neuroscience.

The power iteration method is a foundational algorithm for approximating the dominant eigenvalue and eigenvector of matrices, widely applied in large-scale computations such as Google’s PageRank. While its convergence is well-understood for symmetric matrices, asymmetric (non-Hermitian) matrices present more complex challenges, influenced by the spectral radius and matrix structure. This paper investigates the mathematical relationship between the spectral radius and the convergence rate of power iteration in asymmetric matrices, focusing on both diagonalizable and non-diagonalizable cases. In diagonalizable matrices, convergence depends exponentially on the ratio r=|λ_2 |/|λ_1 |, where λ_1 and λ_2 are the dominant and subdominant eigenvalues. For non-diagonalizable matrices with Jordan blocks, additional polynomial factors k^(m-1) (where m is the block size) slow convergence, even with favorable eigenvalue gaps. Theoretical derivations, including matrix power computations in Jordan form, illustrate these effects. To address limitations such as slow convergence and inefficient information utilization, this paper extends the analysis to advanced iterative methods. Krylov subspace methods overcome single-vector restrictions by constructing optimal approximations in growing subspaces. The Arnoldi process generates orthogonal bases for non-symmetric matrices, leading to Hessenberg projections and Ritz approximations. Shift-and-invert techniques, based on spectral transformation theorems, accelerate convergence and target interior eigenvalues by manipulating the spectrum. Rayleigh quotient iteration introduces adaptive shifting, achieving cubic convergence for symmetric cases and quadratic for asymmetric. Subspace iteration generalizes to multiple eigenvalues using QR decomposition for block processing. Numerical experiments in Python compare basic power iteration with these extensions, demonstrating superior convergence in challenging scenarios (e.g., near-unity ratios or large Jordan blocks). Results highlight how these methods expand applicability in fields like Markov chains and control theory. This study underscores the need for structural awareness in eigenvalue computations and provides a framework for selecting appropriate extensions, enhancing efficiency in practical asymmetric matrix problems.

Phased array technology can be realized with directional control with fixed beam steering. However, its directionally dependent beam pattern limits the efficiency of suppressing undesirable distance interference. This paper presents a guided wave frequency diverse array-based damage location method for thin-walled structures. Firstly, a guided wave frequency diverse array signal model is derived with a relatively small frequency increment that can achieve distance–direction two-dimensional focusing. Secondly, three types of receiving arrays, including a monostatic array, following array, and symmetric array, are constructed to achieve the maximum damage-induced signal amplitude. Finally, a two-dimensional multiple signal classification (MUSIC)-based damage location method is applied for damage imaging in thin-walled structures. Simulations on an aluminum plate and the experiments on an epoxy laminate plate demonstrate the validity and effectiveness of the proposed method.

Multilayer networks are increasingly used in security systems engineering to represent distinct domains of protection, such as physical, digital, human, and infrastructure layers. Each layer is an undirected network, depicted as a symmetric matrix, where nodes correspond to entities and cell values denote their associations across different contexts. This article introduces a Bayesian supervised learning framework for predicting continuous outcomes from multilayer network predictors. Unlike existing methods, it leverages both inter- and intra-layer dependencies using low-rank coefficient models shared across layers. A structured variable selection prior enables identification of influential nodes and edges while maintaining computational efficiency. We demonstrate the framework on Sandia National Laboratories security network data, accurately predicting time to threat detection and highlighting statistically significant nodes. Empirical results show our method outperforms existing approaches in inference and prediction. Supplementary material provides additional simulations, Gibbs sampler construction, and posterior convergence analyses.

This paper determines fundamental operations of a soft matrices and examines the properties of their multiplication. We define and examine the concepts of transpose and symmetric soft square matrices, as well as the notions of determinant and adjoint, illustrating their properties with examples. Additionally, we explore the singular and non-singular forms of soft square matrices and demonstrate them through examples. Finally, we determine the applications of adjoint of square soft matrices in decision making problems.

We study a stochastic differential model for the dynamics of institutional corruption, extending a deterministic three-variable system—corruption perception, proportion of sanctioned acts, and policy laxity—by incorporating Gaussian perturbations into key parameters. We prove global existence and uniqueness of solutions in the physically relevant domain, and we analyze the linearization around the asymptotically stable equilibrium of the deterministic system. Explicit mean square bounds for the linearized process are derived in terms of the spectral properties of a symmetric matrix, providing insight into the temporal validity of the linear approximation. To investigate global behavior, we relate the first exit time from the domain of interest to backward Kolmogorov equations and numerically solve the associated elliptic and parabolic PDEs with FreeFEM, obtaining estimates of expectations and survival probabilities. An application to the case of Mexico highlights nontrivial effects: while the spectral structure governs local stability, institutional volatility can non-monotonically accelerate global exit, showing that highly reactive interventions without effective sanctions increase uncertainty. Policy implications and possible extensions are discussed.

The ever-increasing global demand for low-carbon energy underscores the urgency of water energy harvesting. Despite intensive progress, achieving continuous and efficient water energy harvesting-particularly from abundant, distributed, and low-frequency water flows such as rain, streams, and rivers-remains a critical challenge. Herein, inspired by the classical waterwheel that spatially decouples the gravitational force of flowing water into orthogonal directions for continuous rotation, we report a hybrid, rotatable flowing water-based energy generator (R-FEG) capable of continuous and efficient water energy harvesting at both low and high frequencies. The R-FEG device consists of transistor-like multilayer blades to harvest the kinetic energy of water at the liquid-solid interface via the bulk effect which is favorable at low frequency, and a magnetic rotor on a symmetrical blade array to harvest rotational energy via the electromagnetic effect at high frequency. As a result, the R-FEG device enables self-sustained operation in a wide range of flow rates, collectively delivering an enhanced power of 1131.3 μW at a typical flow rate of 2.0 L min-1. Moreover, the R-FEG exhibits potential versatility as a battery-independent power solution for environmental sensing and outdoor electronics by harvesting water energy across fluctuating flow regimes. This work provides a prospective prototype for water flow energy harvesting, paving a new avenue for scalable, maintenance-free power solutions for applications in remote, offshore, and distributed water energy harvesting.

Let [Formula: see text] be an oriented graph with [Formula: see text] vertices and [Formula: see text] arcs having underlying graph [Formula: see text]. The skew matrix of [Formula: see text] denoted by [Formula: see text] is a [Formula: see text]-skew symmetric matrix. The skew eigenvalues of [Formula: see text] are the eigenvalues of [Formula: see text] and its characteristic polynomial is the skew characteristic polynomial of [Formula: see text]. The sum of the absolute values of the skew eigenvalues is the skew energy of [Formula: see text] and is denoted by [Formula: see text]. In this paper, we study the skew characteristic polynomial, skew eigenvalues of joined union of mixture of oriented bipartite graphs and Eulerian oriented graphs. As a special case we obtain the skew eigenvalues of join of a bipartite oriented graph and the Eulerian oriented graph. Some examples of orientations of well-known graphs are presented to highlight the importance of the results. As applications to our result, we obtain some new infinite families of skew equienergetic oriented graphs. Our results extend and generalize some of the results obtained in C. Adiga and B. R. Rakshith, More skew-equienergetic digraphs, Commun. Comb. Optim. 1(1) (2016) 55–71.

We consider symmetric copositive matrices $A\in \mathbf{M}_n\,(\mathbb{R})$, which by definition satisfy $x^TAx\geq 0$ for all nonzero $x\geq 0$. We introduce the notion the copositive range of a copositive matrix $A$,$$CR(A)=\{x^TAx \,:\, x\geq 0,\; \|x\|_2=1\},$$ and prove that $CR(A)$ is an interval contained in the numerical range of $A$. We focus on the properties and the endpoints of $CR(A)$, which are associated with the Pareto eigenvalues of $A$.

Increasing terrorist activities globally have attracted the attention of many researchers, policy makers and security agencies towards counterterrorism. The clandestine nature of terrorist networks have made them difficult for detection. Existing works have failed to explore computational characterization to design an efficient threat-mining surveillance system. In this paper, a computationally-aware surveillance robot that auto-generates threat information, and transmit same to the cloud-analytics engine is developed. The system offers hidden intelligence to security agencies without any form of interception by terrorist elements. A miniaturized surveillance robot with Hidden Markov Model (MSRHMM) for terrorist computational dissection is then derived. Also, the computational framework for MERHMM is discussed while showing the adjacency matrix of terrorist network as a determinant factor for its operation. The model indicates that the terrorist network have a property of symmetric adjacency matrix while the social network have both asymmetric and symmetric adjacency matrix. Similarly, the characteristic determinant of adjacency matrix as an important operator for terrorist network is computed to be -1 while that of a symmetric and an asymmetric in social network is 0 and 1 respectively. In conclusion, it was observed that the unique properties of terrorist networks such as symmetric and idempotent property conferred a special protection for the terrorist network resilience. Computational robotics is shown to have the capability of utilizing the hidden intelligence in attack prediction of terrorist elements. This concept is expected to contribute in national security challenges, defense and military intelligence.

ABSTRACTApproximate block diagonalization is a problem of transforming a given symmetric matrix as close to block diagonal as possible by symmetric permutations of its rows and columns. This problem arises as a preprocessing stage of various scientific calculations and has been shown to be NP‐complete. In this paper, we consider solving this problem approximately using the D‐Wave Advantage quantum annealer. For this purpose, several steps are needed. First, we have to reformulate the problem as a quadratic unconstrained binary optimization (QUBO) problem. Second, the QUBO has to be embedded into the physical qubit network of the quantum annealer. Third, and optionally, reverse annealing for improving the solution can be applied. We propose two QUBO formulations and four embedding strategies for the problem and discuss their advantages and disadvantages. Through numerical experiments, it is shown that the combination of domain‐wall encoding and D‐Wave's automatic embedding is the most efficient in terms of usage of physical qubits, while the combination of one‐hot encoding and automatic embedding is superior in terms of the probability of obtaining a feasible solution. It is also shown that reverse annealing is effective in improving the solution for medium‐sized problems.

The classical Capelli identity is an important determinantal identity of a matrix with noncommutative entries that determines the center of the enveloping algebra of the general linear Lie algebra, and was used by Weyl as a main tool to study irreducible representations in his famous book on classical groups. In 1996 Okounkov found higher Capelli identities involving immanants of the generating matrix of U ( g l ( n ) ) U(\mathfrak {gl}(n)) which correspond to arbitrary orthogonal idempotent of the symmetric group. It turns out that Williamson also discovered a general Capelli identity of immanants for U ( g l ( n ) ) U(\mathfrak {gl}(n)) in 1981. In this paper, we use a new method to derive a family of even more general Capelli identities that include the aforementioned Capelli identities as special cases as well as many other Capelli-type identities as corollaries. In particular, we obtain generalized Turnbull’s identities for both symmetric and antisymmetric matrices, as well as the generalized Howe-Umeda-Kostant-Sahi identities for antisymmetric matrices which confirm the conjecture of Caracciolo, Sokal, and Sportiello [Electron. J. Combin. 16 (2009), Research Paper 103, 43].

ABSTRACT Given a sequence of symmetric matrices , and a margin , we investigate whether it is possible to find signs such that the operator norm of the signed sum satisfies . Kunisky and Zhang (2023) recently introduced a random version of this problem, where the matrices are drawn from the Gaussian orthogonal ensemble. This model can be seen as a random variant of the celebrated Matrix Spencer conjecture and as a matrix‐valued analog of the symmetric binary perceptron (SBP) in statistical physics. In this work, we establish a satisfiability transition in this problem as with . Our main results are twofold. First, we prove that the expected number of solutions with margin has a sharp threshold at a critical : for the problem is typically unsatisfiable, while for the average number of solutions becomes exponentially large. Second, combining a second‐moment method with recent results from Altschuler (2023) on margin concentration in perceptron‐type problems, we identify a second threshold , such that for the problem admits solutions with high probability. In particular, we establish that a system of Gaussian random matrices can be balanced so that the spectrum of the resulting matrix macroscopically shrinks compared to the typical semicircle law. Finally, under a technical assumption, we show that there exist values of for which the number of solutions has large variance, implying the failure of the second‐moment method and uncovering a richer picture than in the vector‐analog SBP problem. Our proofs rely on concentration inequalities and large deviation properties for the law of correlated Gaussian matrices under spectral norm constraints.

A partially coherent rectangular symmetric array beam for the light source of the laser display system is proposed. The speckle of the laser display system is simulated theoretically by using the angular spectrum method combined with the random phase screen, and the experimental results are given. It is found that unlike traditional lasers, with the increase of the brightness of the laser display system, the speckle contrast does not increase significantly, i.e., the image quality does not decline. The purpose of increasing the brightness of the laser display system without affecting the image quality is achieved by either increasing the number of arrays or increasing the output of laser. The array offset does not affect speckle contrast. Similar to the traditional partially coherent beam, the speckle of the laser display system can be effectively suppressed by reducing the spatial coherence of the partially coherent array beam while keeping the number of arrays unchanged. As a light source, partially coherent array beams provide a novel means to resolve the contradiction between the increase in brightness leading to an increase in speckle contrast and affecting imaging quality, which is different from the traditional methods, and will have a good application prospect in large-size laser display systems or outdoor laser display systems.

Estimating the largest eigenvalue of large symmetric matrices is a critical problem in numerical linear algebra, with wide-ranging applications in science and engineering. Classical iterative methods, such as the Power method, are often computationally expensive or impractical for high-dimensional problems. To address this, we investigate two stochastic methods, Power Monte Carlo (PMC) and Power Quasi-Monte Carlo (PQMC), which integrate Markov chain-based simulations with classical Power iterations to estimate the maximum eigenvalue. These methods are designed to manage both systematic error (from iteration truncation) and stochastic error (from probabilistic sampling), enabling efficient eigenvalue computation even for large matrices. A main contribution of this study is the precise description of the methodology for constructing Almost Optimal PMC and PQMC algorithms, which can cover a broad class of symmetric matrices, including both sparse and dense structures. The algorithms employ a special choice of transition density matrices in constructing the Markov chain, which leads to a significantly reduced variance. Numerical experiments demonstrate that these algorithms outperform classical PMC/PQMC approaches in accuracy and computational complexity, particularly when using robust random number generators and low-discrepancy sequences. They demonstrate that it is vital to find the right balance between the number of steps in the Markov chain and the number of its simulations. Our findings provide new insights into the design of efficient stochastic algorithms for high-dimensional matrix problems and establish a foundation for further optimization and scalability.