Abstract: In this note, we introduce the singular value decomposition Geršgorin set, Γ (A), of an N ×N complex matrix A, where N ≤ ∞. For N finite, the set Γ (A) is similar to the standard Geršgorin set, Γ(A), in that it is a union of N closed disks in the complex plane and it contains the spectrum, σ(A), of A. However, Γ (A) is constructed using column sums of singular value decomposition matr...

∀ ` σ` ∈ R, σ` ≥ 0 (2) ∀ `, `′ 〈u`, u`′〉 = 〈v`, v`′〉 = δ(`, `′) (3) To prove this consider the matrix AA ∈ Rm×m. Set u` to be the `’th eigenvector of AA . By definition we have that AAu` = λ`u`. Since AA T is positive semidefinite we have λ` ≥ 0. Since AA is symmetric we have that ∀ `, `′ 〈u`, u`′〉 = δ(`, `′). Set σ` = √ λ` and v` = 1 σ` Au`. Now we can compute the following: 〈v`, v`′〉 = 1 σ2 `...

We discuss a multilinear generalization of the singular value decomposition. There is a strong analogy between several properties of the matrix and the higher-order tensor decomposition; uniqueness, link with the matrix eigenvalue decomposition, first-order perturbation effects, etc., are analyzed. We investigate how tensor symmetries affect the decomposition and propose a multilinear generaliz...

In this work we consider computing a smooth path for a (block) singular value decomposition of a full rank matrix valued function. We give new theoretical results and then introduce and implement several algorithms to compute a smooth path. We illustrate performance of the algorithms with a few numerical examples.

The singular value decomposition SVD is a powerful technique in many matrix computa tions and analyses Using the SVD of a matrix in computations rather than the original matrix has the advantage of being more robust to numerical error Additionally the SVD exposes the geometric structure of a matrix an important aspect of many matrix calcula tions A matrix can be described as a tranformation fro...

We derive coupled on-line learning rules for the singular value decomposition (SVD) of a cross-covariance matrix. In coupled SVD rules, the singular value is estimated alongside the singular vectors, and the effective learning rates for the singular vector rules are influenced by the singular value estimates. In addition, we use a first-order approximation of Gram-Schmidt orthonormalization as ...

In this paper, a steganography technique for JPEG2000 compressed images using singular value decomposition in wavelet transform domain is proposed. In this technique, DWT is applied on the cover image to get wavelet coefficients and SVD is applied on these wavelet coefficients to get the singular values. Then secret data is embedded into these singular values using scaling factor. Different com...

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 study singular values of a matrix whose one entry varies while all other entries are prescribed. In particular, we find the possible pth singular value of such a matrix, and we define explicitly the unknown entry such that the completed matrix has the minimal possible pth singular value. This in turn determines possible pth singular value of a matrix under rank one perturbation...

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...