We demonstrate that an algorithm proposed by Drineas et. al. in [7] to approximate the singular vectors/values of a matrix A, is not only of theoretical interest but also a fast, viable alternative to traditional algorithms. The algorithm samples a small number of rows (or columns) of the matrix, scales them appropriately to form a small matrix S and computes the singular value decomposition (S...

Given a regular matrix pencil A + μE, we consider the problem of determining the nearest singular matrix pencil with respect to the Frobenius norm. We present new approaches based on the solution of matrix differential equations for determining the nearest singular pencil A+∆A+ μ(E +∆E), one approach for general singular pencils and another one such that A+∆A and E + ∆E have a common left/right...

T. Chan has noted that, even when the singular value decomposition of a matrix A is known, it is still not obvious how to find a rank-revealing QR factorization (RRQR) of A if A has numerical rank deficiency. This paper offers a constructive proof of the existence of the RRQR factorization of any matrix A of size m x n with numerical rank r. The bounds derived in this paper that guarantee the e...

Singular value decomposition is a widely used tool for dimension reduction in multivariate analysis. However, when used for statistical estimation in high-dimensional low rank matrix models, singular vectors of the noise-corrupted matrix are inconsistent for their counterparts of the true mean matrix. In this talk, we suppose the true singular vectors have sparse representations in a certain ba...

A new approach, in a framework of an eigenstructure method using a Hankel matrix, is developed for sinusoidal signal retrieval in white noise. A closed-form solution for the singular pairs of the matrix is defined in terms of the associated sinusoidal signals and noise. The estimated sinusoidal singular vectors are applied to form the noise-free Hankel matrix. A pattern recognition technique is...

Characterization of singular graphs can be reduced to the non-trivial solutions of a system of linear homogeneous equations Ax = 0 for the 0-1 adjacency matrix A. A graph G is singular of nullity η(G) ≥ 1, if the dimension of the nullspace ker(A) of its adjacency matrix A is η(G). Necessary and sufficient conditions are determined for a graph to be singular in terms of admissible induced subgra...

We consider the improvement in accuracy of latent semantic analysis when a part of speech tagger is used to augment a term/document matrix. We first construct an augmented term/document matrix as input into singular value decomposition (SVD). The singular values then serve as principal components for a cosine projection. The results show that the addition of POS tags can decrease ambiguities si...

Sufficient linear matrix inequalities (LMI) conditions for regularization of discrete-time singular systems are given. Then a new class of regularizing stabilizing controllers is discussed. The proposed controllers are the sum of predictive and memoryless state feedbacks. The predictive controller aims to regularizing the singular system while the memoryless state feedback is designed to stabil...

In this manuscript we reduce the computation of the singular values of a general product/quotient of matrices to the computation of the singular values of an upper triangular semiseparable matrix. Compared to the reduction into a bidiagonal matrix the reduction into semiseparable form exhibits a nested subspace iteration. Hence, when there are large gaps between the singular values, these gaps ...

In this study, a numerical solution of singular nonlinear differential equations, stemming from biology and physiology problems, is proposed. The methodology is based on the shifted Chebyshev polynomials operational matrix of derivative and collocation. To assess the accuracy of the method, five numerical problems, such as the human head, Oxygen diffusion and Bessel differential equation, were ...