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Normal Matrices (normal + matrix)

Distribution by Scientific Domains

Mathematics and Statistics	50%

Selected Abstracts

Perturbation of complex polynomials and normal operators

MATHEMATISCHE NACHRICHTEN, Issue 12 2009
Armin Rainer
Abstract We study the regularity of the roots of complex monic polynomials P (t) of fixed degree depending smoothly on a real parameter t. We prove that each continuous parameterization of the roots of a generic C, curve P (t) (which always exists) is locally absolutely continuous. Generic means that no two of the continuously chosen roots meet of infinite order of flatness. Simple examples show that one cannot expect a better regularity than absolute continuity. This result will follow from the proposition that for any t0 there exists a positive integer N such that t , P (t0 ± (t , t0)N) admits smooth parameterizations of its roots near t0. We show that Cn curves P (t) (where n = deg P) admit differentiable roots if and only if the order of contact of the roots is , 1. We give applications to the perturbation theory of normal matrices and unbounded normal operators with compact resolvents and common domain of definition: The eigenvalues and eigenvectors of a generic C, curve of such operators can be arranged locally in an absolutely continuous way (© 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

The perturbation bounds for eigenvalues of normal matrices

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2-3 2005
Wen Li
Abstract In this paper we present some new absolute and relative perturbation bounds of eigenvalues of normal matrices. The bounds depend upon the closeness of perturbed matrices to normal matrices and improve those previous results (Duke Math. J. 1953; 20:37,39, Linear Algebra Appl. 1996; 246:215,223). Copyright © 2004 John Wiley & Sons, Ltd. [source]

Microstructures and adiabatic shear bands formed by ballistic impact in steels and tungsten alloy

FATIGUE & FRACTURE OF ENGINEERING MATERIALS AND STRUCTURES, Issue 12 2003
Z. Q. DUAN
ABSTRACT Projectiles of sintered tungsten alloy were fired directly at two kinds of steel target plates. The microstructures near the perforation of a medium, 0.45% carbon steel target plate can be identified along the radial direction as: melted and rapidly solidified layer, recrystallized fine-grained layer, deformed fine-grained layer, deformed layer and normal matrix. The adiabatic shear bands cannot be found in this intermediate strength steel. The microstructures along the radial direction of perforation of 30CrMnMo steel target plate are different from that of the medium carbon steel. There was a melted and rapidly solidified layer on the surface of the perforation, underneath there was a diffusing layer, and then fine-grained layer appeared as streamlines. Several kinds of adiabatic shear bands were found in this higher strength steel; they had different directions and widths, which were relative to the shock waves, as well as the complex deformation process of penetration. The deformation of the projectiles was rather different when they impacted on target plates of medium carbon steel and 30CrMnMo steel. The projectile that impacted on the medium carbon steel target plate was tamped and its energy dissipated slowly, while that which impacted on the 30CrMnMo steel target plate was sheared and the energy dissipated quickly. [source]

A covariance-adaptive approach for regularized inversion in linear models

GEOPHYSICAL JOURNAL INTERNATIONAL, Issue 2 2007
Christopher Kotsakis
SUMMARY The optimal inversion of a linear model under the presence of additive random noise in the input data is a typical problem in many geodetic and geophysical applications. Various methods have been developed and applied for the solution of this problem, ranging from the classic principle of least-squares (LS) estimation to other more complex inversion techniques such as the Tikhonov,Philips regularization, truncated singular value decomposition, generalized ridge regression, numerical iterative methods (Landweber, conjugate gradient) and others. In this paper, a new type of optimal parameter estimator for the inversion of a linear model is presented. The proposed methodology is based on a linear transformation of the classic LS estimator and it satisfies two basic criteria. First, it provides a solution for the model parameters that is optimally fitted (in an average quadratic sense) to the classic LS parameter solution. Second, it complies with an external user-dependent constraint that specifies a priori the error covariance (CV) matrix of the estimated model parameters. The formulation of this constrained estimator offers a unified framework for the description of many regularization techniques that are systematically used in geodetic inverse problems, particularly for those methods that correspond to an eigenvalue filtering of the ill-conditioned normal matrix in the underlying linear model. Our study lies on the fact that it adds an alternative perspective on the statistical properties and the regularization mechanism of many inversion techniques commonly used in geodesy and geophysics, by interpreting them as a family of ,CV-adaptive' parameter estimators that obey a common optimal criterion and differ only on the pre-selected form of their error CV matrix under a fixed model design. [source]

How to multiply a matrix of normal equations by an arbitrary vector using FFT

ACTA CRYSTALLOGRAPHICA SECTION A, Issue 6 2008
Boris V. Strokopytov
This paper describes a novel algorithm for multiplying a matrix of normal equations by an arbitrary real vector using the fast Fourier transform technique. The algorithm allows full-matrix least-squares refinement of macromolecular structures without explicit calculation of the normal matrix. The resulting equations have been implemented in a new computer program, FMLSQ. A preliminary version of the program has been tested on several protein structures. The consequences for crystallographic refinement of macromolecules are discussed in detail. [source]