This chapter is about eigenvalues and singular values of matrices. Computational algorithms and sensitivity to perturbations are both discussed. An eigenvalue and eigenvector of a square matrix A are a scalar λ and a nonzero vector x so that Ax = λx. A singular value and pair of singular vectors of a square or rectangular matrix A are a nonnegative scalar σ and two nonzero vectors u and v so th...

In recent years, in various sectors of our society, there have been nationalist stirrings which were crystallized and articulated by the late Claro M. Recto, There were jealous demands for the recognition of Philippine sovereignty on the Bases question. There were appeals for the correction of the iniquitous economic relations between the Philippines and the United States. For a time, Filipino ...

This article presents a new subspace-based technique for reducing the noise of signals in time-series. In the proposed approach, the signal is initially represented as a data matrix. Then using Singular Value Decomposition (SVD), noisy data matrix is divided into signal subspace and noise subspace. In this subspace division, each derivative of the singular values with respect to rank order is u...

In this paper we propose three robust and semi-blind digital color image watermarking algorithms. These algorithms are based on hybrid transforms using the combination of Discrete Cosine Transform (DCT) and Singular Value Decomposition (SVD), Discrete Wavelet Transform (DWT) and Singular Value Decomposition (SVD), Discrete Wavelet Transform (DWT), Discrete Cosine Transform (DCT) and Singular Va...

A QR–method for computing the singular values via semiseparable matrices. Abstract The standard procedure to compute the singular value decomposition of a dense matrix, first reduces it into a bidiagonal one by means of orthogonal transformations. Once the bidiagonal matrix has been computed, the QR–method is applied to reduce the latter matrix into a diagonal one. In this paper we propose a ne...

1.1 Singular values and matrix inversion For non-symmetric matrices, the eigenvalues and singular values are not equivalent. However, they share one important property: Fact 1 A matrix A, NxN, is invertible iff all of its singular values are non-zero.

Accurately computing the singular values of long products of matrices is important for estimating Lyapunov exponents: λi = limn→∞(1/n) log σi(An · · ·A1). Algorithms for computing singular values of products in fact compute the singular values of a perturbed product (An +En) · · · (A1 +E1). The question is how small are the relative errors of the singular values of the product with respect to t...

In this paper, we develop a new image quality assessment scheme with the use of Singular Value Decomposition (SVD). In a previous work [1], only the singular values out of SVD were used for quality assessment. We show that singular vectors are even more important than singular values for quality assessment. Hence, we use both singular vectors and values for comprehensive image quality evaluatio...