scholarly journals Random non-Abelian G-circulant matrices. Spectrum of random convolution operators on large finite groups

2020 ◽  
pp. 2250002
Author(s):  
Radosław Adamczak
Keyword(s):  
Symmetric Groups ◽  
Singular Values ◽  
Singular Value ◽  
Circulant Matrices ◽  
Plancherel Measure ◽  
Convolution Operators ◽  
Limiting Distributions ◽  
Dihedral Groups ◽  
Limiting Behavior ◽  
Abelian Case

We analyze the limiting behavior of the eigenvalue and singular value distribution for random convolution operators on large (not necessarily Abelian) groups, extending the results by Meckes for the Abelian case. We show that for regular sequences of groups, the limiting distribution of eigenvalues (respectively singular values) is a mixture of eigenvalue (respectively singular value) distributions of Ginibre matrices with the directing measure being related to the limiting behavior of the Plancherel measure of the sequence of groups. In particular, for the sequence of symmetric groups, the limiting distributions are just the circular and quarter circular laws, whereas e.g. for the dihedral groups, the limiting distributions have unbounded supports but are different than in the Abelian case. We also prove that under additional assumptions on the sequence of groups (in particular, for symmetric groups of increasing order) families of stochastically independent random convolution operators converge in moments to free circular elements. Finally, in the Gaussian case, we provide Central Limit Theorems for linear eigenvalue statistics.

scholarly journals On singular values distribution of a matrix large auto-covariance in the ultra-dimensional regime

2015 ◽  
Vol 04 (04) ◽  
pp. 1550015
Author(s):  
Qinwen Wang ◽  
Jianfeng Yao
Keyword(s):  
Singular Values ◽  
Singular Value ◽  
Moment Condition ◽  
Fourth Moment ◽  
Limiting Behavior ◽  
The Moment ◽  
Fixed Positive Integer ◽  
The Right ◽  
Auto Covariance ◽  
Random Limit

Let [Formula: see text] be a sequence of independent real random vectors of [Formula: see text]-dimension and let [Formula: see text] be the lag-[Formula: see text] ([Formula: see text] is a fixed positive integer) auto-covariance matrix of [Formula: see text]. This paper investigates the limiting behavior of the singular values of [Formula: see text] under the so-called ultra-dimensional regime where [Formula: see text] and [Formula: see text] in a related way such that [Formula: see text]. First, we show that the singular value distribution of [Formula: see text], after a suitable normalization, converges to a non-random limit [Formula: see text] (quarter law) under the fourth-moment condition. Second, we establish the convergence of its largest singular value to the right edge of the support of [Formula: see text]. Both results are derived using the moment method.

scholarly journals Exact relation between singular value and eigenvalue statistics

2016 ◽  
Vol 05 (04) ◽  
pp. 1650015 ◽  
Author(s):  
Mario Kieburg ◽  
Holger Kösters
Keyword(s):  
Singular Values ◽  
Singular Value ◽  
Specific Class ◽  
Rotation Invariant ◽  
Exact Relation ◽  
Finite Matrix ◽  
Eigenvalue Statistics ◽  
Probability Densities ◽  
Analytical Relations ◽  
Matrix Spaces

We use classical results from harmonic analysis on matrix spaces to investigate the relation between the joint densities of the singular values and the eigenvalues for complex random matrices which are bi-unitarily invariant (also known as isotropic or unitary rotation invariant). We prove that each of these joint densities determines the other one. Moreover, we construct an explicit formula relating both joint densities at finite matrix dimension. This relation covers probability densities as well as signed densities. With the help of this relation we derive general analytical relations among the corresponding kernels and biorthogonal functions for a specific class of polynomial ensembles. Furthermore, we show how to generalize the relation between the singular value and eigenvalue statistics to certain situations when the ensemble is deformed by a term which breaks the bi-unitary invariance.

scholarly journals Computing mu-values for Representations of Symmetric Groups in Engineering Systems

2018 ◽  
Vol 11 (3) ◽  
pp. 774-792
Author(s):  
Mutti-Ur Rehman ◽  
M. Fazeel Anwar
Keyword(s):  
Control Systems ◽  
Symmetric Groups ◽  
Singular Values ◽  
Linear Control ◽  
Linear Control Systems ◽  
Complex Numbers ◽  
Engineering Systems ◽  
Matrix Representations ◽  
The Matrix

In this article we consider the matrix representations of finite symmetric groups Sn over the filed of complex numbers. These groups and their representations also appear as symmetries of certain linear control systems [5]. We compute the structure singular values (SSV) of the matrices arising from these representations. The obtained results of SSV are compared with well-known MATLAB routine mussv.

scholarly journals Through-Wall Image Enhancement Based on Singular Value Decomposition

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Muhammad Mohsin Riaz ◽  
Abdul Ghafoor
Keyword(s):  
Singular Value Decomposition ◽  
Image Enhancement ◽  
Visual Inspection ◽  
Signal To Noise Ratio ◽  
Singular Values ◽  
Singular Value ◽  
Information Theoretic ◽  
Value Decomposition ◽  
Suppress Noise ◽  
Wall Imaging

Singular value decomposition and information theoretic criterion-based image enhancement is proposed for through-wall imaging. The scheme is capable of discriminating target, clutter, and noise subspaces. Information theoretic criterion is used with conventional singular value decomposition to find number of target singular values. Furthermore, wavelet transform-based denoising is performed (to further suppress noise signals) by estimating noise variance. Proposed scheme works also for extracting multiple targets in heavy cluttered through-wall images. Simulation results are compared on the basis of mean square error, peak signal to noise ratio, and visual inspection.

Lyapunov and marginal instability in Hamiltonian systems

1999 ◽  
Vol 77 (8) ◽  
pp. 603-633 ◽  
Author(s):  
J Grindlay
Keyword(s):  
Lyapunov Exponents ◽  
Nonlinear Oscillator ◽  
Linear Chain ◽  
Mathieu Equation ◽  
Singular Values ◽  
Space Velocity ◽  
Singular Value ◽  
The Stability ◽  
Value Decomposition ◽  
Simple Systems

The variational equations and the evolution matrix are introduced and used to discuss the stability of a bound Hamiltonian trajectory. Singular-value decomposition is applied to the evolution matrix. Singular values and Lyapunov exponents are defined and their properties described. The singular-value expansion of the phase-space velocity is derived. Singular values and Lyapunov exponents are used to characterize the stability behaviour of five simple systems, namely, the nonlinear oscillator with cubic anharmonicity, the quasi-periodic Mathieu equation, the Hénon-Heilesmodel, the 4+2 linear chain with cubic anharmonicity, and an integrable system of arbitrary order.PACS Nos.: 03.20, 05.20

Automorphism groups of simply reducible groups

2020 ◽  
pp. 2150088
Author(s):  
Yongzhi Luan
Keyword(s):  
Computer Algebra System ◽  
Coxeter Groups ◽  
Eigenvalue Problems ◽  
Automorphism Groups ◽  
Symmetric Groups ◽  
Molecular Symmetry ◽  
Dihedral Groups ◽  
Inner Automorphism ◽  
The Moment

Simply reducible groups are closely related to the eigenvalue problems in quantum theory and molecular symmetry in chemistry. Classification of simply reducible groups is still an open problem which is interesting to physicists. Since there are not many examples of simply reducible groups in literature at the moment, we try to find some examples of simply reducible groups as candidates for the classification. By studying the automorphism and inner automorphism groups of symmetric groups, dihedral groups, Clifford groups and Coxeter groups, we find some new examples of candidates. We use the computer algebra system GAP to get most of these automorphism and inner automorphism groups.

scholarly journals Toward efficacy of piecewise polynomial truncated singular value decomposition algorithm in moving force identification

2019 ◽  
Vol 22 (12) ◽  
pp. 2687-2698 ◽  
Author(s):  
Zhen Chen ◽  
Lifeng Qin ◽  
Shunbo Zhao ◽  
Tommy HT Chan ◽  
Andy Nguyen
Keyword(s):  
Singular Value Decomposition ◽  
Singular Values ◽  
Singular Value ◽  
Decomposition Algorithm ◽  
Identification Accuracy ◽  
Piecewise Polynomial ◽  
Truncated Singular Value Decomposition ◽  
Force Identification ◽  
Moving Force ◽  
Value Decomposition

This article introduces and evaluates the piecewise polynomial truncated singular value decomposition algorithm toward an effective use for moving force identification. Suffering from numerical non-uniqueness and noise disturbance, the moving force identification is known to be associated with ill-posedness. An important method for solving this problem is the truncated singular value decomposition algorithm, but the truncated small singular values removed by truncated singular value decomposition may contain some useful information. The piecewise polynomial truncated singular value decomposition algorithm extracts the useful responses from truncated small singular values and superposes it into the solution of truncated singular value decomposition, which can be useful in moving force identification. In this article, a comprehensive numerical simulation is set up to evaluate piecewise polynomial truncated singular value decomposition, and compare this technique against truncated singular value decomposition and singular value decomposition. Numerically simulated data are processed to validate the novel method, which show that regularization matrix [Formula: see text] and truncating point [Formula: see text] are the two most important governing factors affecting identification accuracy and ill-posedness immunity of piecewise polynomial truncated singular value decomposition.

Singular value decomposition for cross‐well tomography

Geophysics ◽  
1993 ◽  
Vol 58 (11) ◽  
pp. 1655-1661 ◽  
Author(s):  
Reinaldo J. Michelena
Keyword(s):  
Singular Value Decomposition ◽  
Null Space ◽  
Vertical Component ◽  
Singular Values ◽  
Singular Value ◽  
Velocity Estimation ◽  
One Dimensional ◽  
Anisotropic Models ◽  
Value Decomposition ◽  
Frequency Variations

I perform singular value decomposition (SVD) on the matrices that result in tomographic velocity estimation from cross‐well traveltimes in isotropic and anisotropic media. The slowness model is parameterized in four ways: One‐dimensional (1-D) isotropic, 1-D anisotropic, two‐dimensional (2-D) isotropic, and 2-D anisotropic. The singular value distribution is different for the different parameterizations. One‐dimensional isotropic models can be resolved well but the resolution of the data is poor. One‐dimensional anisotropic models can also be resolved well except for some variations in the vertical component of the slowness that are not sensitive to the data. In 2-D isotropic models, “pure” lateral variations are not sensitive to the data, and when anisotropy is introduced, the result is that the horizontal and vertical component of the slowness cannot be estimated with the same spatial resolution because the null space is mostly related to horizontal and high frequency variations in the vertical component of the slowness. Since the distribution of singular values varies depending on the parametrization used, the effect of conventional regularization procedures in the final solution may also vary. When the model is isotropic, regularization translates into smoothness, and when the model is anisotropic regularization not only smooths but may also alter the anisotropy in the solution.

Sign in / Sign up

Export Citation Format

Share Document