We calculate explicitly the singular vectors of the Virasoro algebra with the central charge c ≤ 1. As a result, we have an infinite sequence of the singular vectors for each Fock space with given central charge and highest weight, and all its elements can be written in terms of the Jack symmetric functions with rectangular Young diagram. Since the paper of Belavin, Polyakov and Zamolodchikov [...

abstract The singular value decomposition has a number of applications in digital signal processing. However, the the decomposition must be computed from a matrix consisting of both signal and noise. It is therefore important to be able to assess the eeects of the noise on the singular values and singular vectors | a problem in classical perturbation theory. In this paper we survey the perturba...

The present paper considers singular value decomposition (SVD) for a class of linear time-varying systems. The class considered herein describes timedriven switched linear systems. Based on an appropriate input-output description, the calculation method of singular values and singular vectors is derived. The SVD enables us to characterize the dominant input–output signals using singular vectors...

carries important information about the structure of the data set, especially when the rank k of X is small. In particular, the columns of U (known as the left singular vectors of X) span the principal directions of the data set and can be used as basis vectors for building up typical signals, and the diagonal entries of Σ (known as the singular values of X) reflect the energy of the data set i...

The harmonic Lanczos bidiagonalization method can be used to compute the smallest singular triplets of a large matrix A. We prove that for good enough projection subspaces harmonic Ritz values converge if the columns of A are strongly linearly independent. On the other hand, harmonic Ritz values may miss some desired singular values when the columns of A are almost linearly dependent. Furthermo...

Spectral embedding based on the Singular Value Decomposition (SVD) is a widely used “preprocessing” step in many learning tasks, typically leading to dimensionality reduction by projecting onto a number of dominant singular vectors and rescaling the coordinate axes (by a predefined function of the singular value). However, the number of such vectors required to capture problem structure grows w...

An efficient Singular Value Decomposition (SVD) algorithm is an important tool for distributed and streaming computation in big data problems. It is observed that update of singular vectors of a rank-1 perturbed matrix is similar to a Cauchy matrix-vector product. With this observation, in this paper, we present an efficient method for updating Singular Value Decomposition of rank1 perturbed ma...