Singular Eigenvalues Research Articles

On the mathematics of: “Criticality and the fractal structure of −5/3 turbulent cascades”

The linearization to DRM model performed in Cabrera et al. (2021), which has the intend of establishing a relationship between succession of singular matrix eigenvalues of such equation and the cascades of energy in Kolmogorov’s −5/3 turbulence model, is examined. Their report proposes a small group of particular solutions as a dense populating ensemble of the general dynamics. However this assumption drives to some mathematical inconsistencies. This note points some of them out.

Random matrices associated to Young diagrams

We consider the singular values of certain Young diagram shaped random matrices. For block-shaped random matrices, the empirical distribution of the squares of the singular eigenvalues converges almost surely to a distribution whose moments are a generalization of the Catalan numbers. The limiting distribution is the density of a product of rescaled independent Beta random variables and its Stieltjes–Cauchy transform has a hypergeometric representation. In special cases we recover the Marchenko–Pastur and Dykema–Haagerup measures of square and triangular random matrices, respectively. We find a further factorization of the moments in terms of two complex-valued random variables that generalizes the factorization of the Marchenko–Pastur law as product of independent uniform and arcsine random variables.

Parameters identification of closely spaced modes with the covariance-drivern stochastic subspace and damping ratio dispersion method

Electromechanical equipment with quasi-symmetric and partially symmetrical structures is widespread in the industry, which not only makes the dynamic characteristics of the system more complex but also leads to severe closely spaced modes. The existence of closely spaced modes can lead to more confusion, because severe closely spaced modes have similar frequency eigenvalues and some singular eigenvalues are prone to generate spurious modes in matrix computations. In this paper, the covariance-drivern stochastic subspace and damping ratio dispersion (SSI-COV-DRD) method is proposed, which in comparison with previous methods, identification process directly is performed in the stochastic subspace by solving the system matrix of the time domain signal to overcome modal aliasing of the frequency-based method. Firstly, a Toeplitz matrix is constructed with a time series signal and then the system modal parameters are gained from the system matrix after performing singular value decomposition (SVD) to the Toeplitz matrix. Subsequently, the poles of the continuous model order are classified by a hierarchical clustering algorithm to obtain stabilization diagram and the real physical modes are automatically extracted from the stabilization diagram according to the choice criteria. Finally, the damping ratio dispersion index is calculated to express the nonlinear distribution characteristic of the damping ratio. Numerical examples for the five-order free vibration simulation signals with noise-free and noise indicate that the proposed method can simultaneously identify the modal frequency and damping ratio with high accuracy in severe closely spaced modes, especially for the repeated frequency modes. Since the bolt connection rotor of the aero-engine usually produces severe closely spaced modes in vibration due to the structural symmetry, the frequency and damping ratio are extracted and the damping ratio dispersion index is calculated from different bolt loosening states, which is helpful for the bolt loosening detection of aviation safety assurance and maintenance.

Comparative performance of novel nodal-to-edge finite elements over conventional nodal element for electromagnetic analysis

In nodal-based finite element method (FEM), degrees of freedom are associated with the nodes of the element whereas, for edge FEM, degrees of freedom are assigned to the edges of the element. Edge element is constructed based on Whitney spaces. Nodal elements impose both tangential and normal continuity of vector or scalar fields across interface boundaries. But in edge elements, only tangential continuity is imposed across interface boundaries, which is consistent with electromagnetic field problems. Therefore the required continuities in the electromagnetic analysis are directly obtained with edge elements whereas in nodal elements they are attained through potential formulations. Hence, while using edge elements, field variables are directly calculated but with nodal elements, post-processing is required to obtain the field variables from the potentials. Here, we present the finite element formulations with the edge element as well as with nodal elements. Thereafter, we have demonstrated the relative performances of different nodal and edge elements through a series of examples. All possible complexities like curved boundaries, non-convex domains, sharp corners, non-homogeneous domains have been addressed in those examples. The robustness of edge elements in predicting the singular eigen values for the domains with sharp edges and corners is evident in the analysis. A better coarse mesh accuracy has been observed for all the edge elements as compared to the respective nodal elements. Edge elements are also not susceptible to mesh distortion.

Estimates of singular numbers (s-numbers) for a class of degenerate elliptic operators

In this paper we study a class of degenerate elliptic equations with an arbitrary power degeneracy on the line. Based on the research carried out in the course of the work, the authors propose methods to overcome various difficulties associated with the behavior of functions from the definition domain for a differential operator with piecewise continuous coefficients in a bounded domain, which affect the spectral characteristics of boundary value problems for degenerate elliptic equations. It is shown the conditions imposed on the coefficients at the lowest terms of the equation, which ensure the existence and uniqueness of the solution. The existence, uniqueness, and smoothness of a solution are proved, and estimates are found for singular numbers (s-numbers) and eigenvalues of the semiperiodic Dirichlet problem for a class of degenerate elliptic equations with arbitrary power degeneration.

Introducing discrepancy values of matrices with application to bounding norms of commutators

We introduce discrepancy values, quantities inspired by the notion of spectral spread of Hermitian matrices. We define them as the discrepancy between two consecutive Ky-Fan-like seminorms. As a result, discrepancy values share many properties with singular values and eigenvalues, yet are substantially different to merit their own study. We describe key properties of discrepancy values, and establish tools such as representation theorems, majorization inequalities, convex formulations, etc., for working with them. As an important application, we illustrate the role of discrepancy values in deriving tight bounds on the norms of commutators.

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 analysis of the perturbation error bounds for singular and eigenvalues of diagonally dominant matrices

In this paper, we investigate the error bounds of singular values and eigenvalues for diagonally dominant Hermite matrices when the matrix on its off-diagonal entries and on diagonally dominant part is perturbed with bound ε. Our results show that not only is the bound independent of the condition number of the matrix, but the matrix itself need not be symmetric positive definite. Furthermore, the bounds are applicable to both the eigenvalue and singular value of the matrix and can be used in generalized eigenvalue problems.

Accurate Spectral Collocation Computations of High Order Eigenvalues for Singular Schrödinger Equations-Revisited

In this paper, we continue to solve as accurately as possible singular eigenvalues problems attached to the Schrödinger equation. We use the conventional ChC and SiC as well as Chebfun. In order to quantify the accuracy of our outcomes, we use the drift with respect to some parameters, i.e., the order of approximation N, the length of integration interval X, or a small parameter ε, of a set of eigenvalues of interest. The deficiency of orthogonality of eigenvectors, which approximate eigenfunctions, is also an indication of the accuracy of the computations. The drift of eigenvalues provides an error estimation and, from that, one can achieve an error control. In both situations, conventional spectral collocation or Chebfun, the computing codes are simple and very efficient. An example for each such code is displayed so that it can be used. An extension to a 2D problem is also considered.

Resampling the transmission matrix in an aberration-corrected Bessel mode basis.

The study of the optical transmission matrix (TM) of a sample reveals important statistics of light transport through it. The accuracy of the statistics depends strongly on the orthogonality and completeness of the basis in which the TM is measured. While conventional experimental methods suffer from sampling effects and optical aberrations, we use a basis of Bessel modes of the first kind to faithfully recover the singular values, eigenvalues and eigenmodes of light propagation through a finite thickness of air.

Numerical methods for accurate computation of the eigenvalues of Hermitian matrices and the singular values of general matrices

This paper offers a review of numerical methods for computation of the eigenvalues of Hermitian matrices and the singular values of general and some classes of structured matrices. The focus is on the main principles behind the methods that guarantee high accuracy even in the cases that are ill-conditioned for the conventional methods. First, it is shown that a particular structure of the errors in a finite precision implementation of an algorithm allows for a much better measure of sensitivity and that computation with high accuracy is possible despite a large classical condition number. Such structured errors incurred by finite precision computation are in some algorithms e.g. entry-wise or column-wise small, which is much better than the usually considered errors that are in general small only when measured in the Frobenius matrix norm. Specially tailored perturbation theory for such structured perturbations of Hermitian matrices guarantees much better bounds for the relative errors in the computed eigenvalues. Secondly, we review an unconventional approach to accurate computation of the singular values and eigenvalues of some notoriously ill-conditioned structured matrices, such as e.g. Cauchy, Vandermonde and Hankel matrices. The distinctive feature of accurate algorithms is using the intrinsic parameters that define such matrices to obtain a non-orthogonal factorization, such as the LDU factorization, and then computing the singular values of the product of thus computed factors. The state of the art software is discussed as well.

ПОВЫШЕНИЕ РАЗРЕШАЮЩЕЙ СПОСОБНОСТИ СЛАБОКОНТРАСТНЫХ ИЗОБРАЖЕНИЙ В ПРОСТРАНСТВЕ ФАЗОВЫХ ХАРАКТЕРИСТИК СИНГУЛЯРНОГО РАЗЛОЖЕНИЯ

Рассмотрены информационные возможности фазовых характеристик, синтезируемых на основе метода сингулярного разложения применительно к решению задач сегментации при визуальном анализе слабоконтрастных изображений. Показано, что переход в новое пространство признаков, полученных на основе параметров ортогонального преобразования, обеспечивает повышение чувствительности и разрешаюшей способности при выявлении объектов интереса (аномальных областей). Приведены экспериментальные результаты на примере реальных изображениях.

Singular Values, Eigenvalues and Diagonal Elements Of The Commutator Of 2x2 Rank One Matrices

We determine the region of singular values of the commutator XY-YX for 2x2 rank one complex matrices X and Y. This answers in affirmative a conjecture raised in [Wenzel 2015] when 2x2 matrices are concerned. Our approach and proofs also lead to a complete relation between the singular values, eigenvalues and diagonal elements of the commutator under consideration.

Some inequalities for eigenvalues and symplectic eigenvalues of positive definite matrices

We show that for any two [Formula: see text] matrices [Formula: see text] and [Formula: see text] we have the inequality [Formula: see text] where [Formula: see text] and [Formula: see text] denote the decreasingly ordered singular values and eigenvalues of [Formula: see text]. As an application, we show that for [Formula: see text] real positive definite matrices the symplectic eigenvalues [Formula: see text] under some special conditions, satisfy the inequality [Formula: see text].

Asymptotic profile and Morse index of nodal radial solutions to the Hénon problem

We compute the Morse index of nodal radial solutions to the H\'enon problem \[\left\{\begin{array}{ll} -\Delta u = |x|^{\alpha}|u|^{p-1} u \qquad & \text{ in } B, \newline u= 0 & \text{ on } \partial B, \end{array} \right. \] where $B$ stands for the unit ball in ${\mathbb R}^N$ in dimension $N\ge 3$, $\alpha>0$ and $p$ is near at the threshold exponent for existence of solutions $p_{\alpha}=\frac{N+2+2\alpha}{N-2}$, obtaining that \begin{align*} m(u_p) & = m \sum\limits_{j=0}^{1+\left[{\alpha}/{2}\right]} N_j \quad & \mbox{ if $\alpha$ is not an even integer, or} \newline m(u_p)& = m\sum\limits_{j=0}^{ \alpha /2} N_j + (m-1) N_{1+\alpha/ 2} & \mbox{ if $\alpha$ is an even number.} \end{align*} Here $N_j$ denotes the multiplicity of the spherical harmonics of order $j$. The computation builds on a characterization of the Morse index by means of a one dimensional singular eigenvalue problem, and is carried out by a detailed picture of the asymptotic behavior of both the solution and the singular eigenvalues and eigenfunctions. In particular it is shown that nodal radial solutions have multiple blow-up at the origin, where each node converges (up to a suitable rescaling) to the bubble shaped solution of a limit problem. As side outcome we see that solutions are nondegenerate for $p$ near at $p_{\alpha}$, and we give an existence result in perturbed balls.

The RESPECT method. Simple proof of finding estimates from below for minimal singular eigenvalues of random matrices whose entries have zero means and bounded variances

Abstract The lower bounds for the minimal singular eigenvalue of the matrix whose entries have zero means and bounded variances are obtained. The new method is based on the G-method of perpendiculars and the RESPECT method.

Complex near-grazing dynamics in impact oscillators

Impact oscillators frequently appear in various physical and engineering systems with non-smooth characteristics and can exhibit different dynamic behavior from the smooth nonlinear systems, including grazing bifurcation in which an impact with zero velocity occurs. This paper investigates the near-grazing dynamics of the multi-degree-of-freedom impact oscillators in the small neighborhood of degenerate grazing points, with a focus on the stability and potential bifurcations of near-grazing period-one impact motions. The high order zero time discontinuity mapping method is applied to perform the prospective analyses of stability and bifurcations. Particularly, this paper shows that the peculiar Neimark-Sacker bifurcations regaining the stability of near-grazing period-one impact motion can be induced by two different ways, either through the interaction between the singular and regular real eigenvalues or via a grazing bifurcation directly. A two degree-of-freedom impact oscillator is taken as an example to present detailed numerical results for the verification of proposed analysis.

Sparse recovery from extreme eigenvalues deviation inequalities

This article provides a new toolbox to derive sparse recovery guarantees – that is referred to as “stable and robust sparse regression” (SRSR) – from deviations on extreme singular values or extreme eigenvalues obtained in Random Matrix Theory. This work is based on Restricted Isometry Constants (RICs) which are a pivotal notion in Compressed Sensing and High-Dimensional Statistics as these constants finely assess how a linear operator is conditioned on the set of sparse vectors and hence how it performs in SRSR. While it is an open problem to construct deterministic matrices with apposite RICs, one can prove that such matrices exist using random matrices models. In this paper, we show upper bounds on RICs for Gaussian and Rademacher matrices using state-of-the-art deviation estimates on their extreme eigenvalues. This allows us to derive a lower bound on the probability of getting SRSR. One benefit of this paper is a direct and explicit derivation of upper bounds on RICs and lower bounds on SRSR from deviations on the extreme eigenvalues given by Random Matrix theory.

Application and adaption of Dynamic Mode Decomposition to experimental and numerical data of turbomachinery flow

AbstractUnsteady turbomachinery flows are challenging in terms of resolving turbulent characteristics and instability mechanisms like rotating stall, rotor‐stator interaction and von Karmán vortex streets. Furthermore, in part load conditions these flow phenoma can induce vibrations leading to resonance problems and high cycle fatigue. Flowfield measurements of a centrifugal fan were taken by high‐speed particle image velocimetry (PIV). CFD results, calculated as unsteady Reynolds averaged Navier Stokes (URANS) and based on Menter's shear‐stress transport (SST) model, are the source of numerical data. In former examinations discrepancies between experiment and simulation in time‐averaged velocity profil data for part load conditions have been stated. The source of theses discrepancies is still unknown. Hence, a new approach is needed for further examination of measured and calculated flowfields. The Dynamic Mode Decomosition is employed in order to compare experimental and numerical datasets. Information about coherent structures and the dynamical behaviour are contained in the eigenvalues and their associated eigenvectors offered by DMD. While the eigenvectors are describing the coherent structures, the associated eigenvalues describe the dynamic behaviour of the structure. The eigenvectors are denoted as DMD‐Modes. The results acquired by DMD based on experimental and numerical data are compared to investigate congruent behaviour and therefore if it is possible to obtain reliable information about dynamic behaviour of a system even with URANS simulations. Comparisons between singular value and eigenvalues are presented and discussed.

Spectral properties of the binary Hadamard matrices

ABSTRACTA binary Hadamard matrix (also called an S-matrix) is an -matrix formed by taking an Hadamard matrix in which the entries in the first row and column are , changing's to 0's and 's to 's, and deleting the first row and column. In this paper, some spectral properties of the binary Hadamard matrices are derived. All singular values and eigenvalues' modulus of an arbitrary S-matrix are obtained. Two special types of binary Hadamard matrices, namely the symmetric and the skew-type ones, are analysed in more detail. In particular, we prove that an S-matrix (regardless of its order n) is of skew type if, and only if, all its eigenvalues different from the largest one (in modulus) are imaginary and have real part 1/2. Finally, the symmetric and skew-symmetric parts of an S-matrix are analysed.