This paper analyzes many weak moment asymptotics under the possibility of similar moments. The possibility of highly related moments arises when there are many of them. Knight and Fu (2000) designate the issue of similar regressors as the “nearly singular” design in the least squares case. In the nearly singular design, the sample variance converges to a singular limit term. However, Knight and...

The computation of a singular value decomposition of an m× n matrix A is certainly one of the most often demanded tasks in various applications. There are many algorithms for computing the full or partial singular value decomposition. Among them, the one-sided Jacobi method (coupled with some orderings) is reputable for its ability to compute the singular values as well as left and right singul...

Two issues concerning the construction of square matrices with prescribed singular values and eigenvalues are addressed. First, a necessary and sufficient condition for the existence of an n × n complex matrix with n given nonnegative numbers as singular values and m(≤ n) given complex numbers to be m of the eigenvalues is determined. This extends the classical result of Weyl and Horn treating ...

The purpose of this study aims at the use of machine vision for automatic surface inspection, in which defects are embedded in homogenously LCDs textured surfaces. The proposed method does not rely on local features of textures. It is based on a global image reconstruction scheme using the singular value decomposition (SVD). The singular values on the decomposed diagonal matrix represent differ...

The set of all n-by-n complex matrices is denoted by Cn×n. Let A aij ∈ Cn×n. Denote the Hermitian adjoint of matrix A by A∗. Then the singular values of A are the eigenvalues of AA∗ . It is well known that matrix singular values play a very key role in theory and practice. The location of singular values is very important in numerical analysis and many other applied fields. For more review abou...

In this paper, we discuss convergence of the extrapolated iterative methods for solving singular linear systems. A general principle of extrapolation is presented. The semiconvergence of an extrapolated method induced by a regular splitting and a nonnegative splitting is proved whenever the coe cient matrix A is a singular M -matrix with ‘property c’ and an irreducible singular M -matrix, respe...

A Typical Numerical Computing Problem To illustrate the two paradigms, we will use MATLAB to test a hypothesis regarding Girko’s circular law. Girko’s law states that the eigenvalues of a random N-by-N matrix whose elements are drawn from a normal distribution tend to lie inside a circle of radius sqrt(N) for large enough N. Our hypothesis is that Girko’s circular law can be modified to apply t...

We highlight a new algorithm for evaluating the surface intersection curve using a matrix formulation. The projection of the intersection curve is represented as the singular set of a bivariate matrix polynomial. The resulting algorithm for evaluating the intersection curve is based on matrix computations like eigendecomposition and singular value decomposition. Furthermore, at each stage of th...

A well–known property of an irreducible singular M–matrix is that it has a generalized inverse which is non–negative, but this is not always true for any generalized inverse. The authors have characterized when the Moore–Penrose inverse of a symmetric, singular, irreducible and tridiagonal M–matrix is itself an M–matrix. We aim here at giving new explicit examples of infinite families of matric...

Disclaimer: These notes have not been subjected to the usual scrutiny reserved for formal publications. In this lecture we describe applications of low rank approximation in optimization. Firstly, let us give a short overview of the last lecture. We defined the operator norm of a matrix ‖.‖2 and the Frobenius norm ‖.‖F and we showed that the best rank k approximation of a given matrix M is the ...