We develop a general method for obtaining bounds for the sensitivity of the solution of the stable Lyapunov equation. Our arguments not only work simultaneously for both the continuous and the discrete time Lyapunov equations, they work equally well for more general equations that have been studied. Further, we obtain bounds valid for all unitarily invariant norms. x 1 Introduction In this note...

As in the multivariate setting, the class of elliptical distributions on separable Hilbert spaces serves as an important vehicle and reference point for the development and evaluation of robust methods in functional data analysis. In this paper, we present a simple characterization of elliptical distributions on separable Hilbert spaces, namely we show that the class of elliptical distributions...

Low-rank approximation of large and/or sparse matrices is important in many applications, and the singular value decomposition (SVD) gives the best low-rank approximations with respect to unitarily-invariant norms. In this paper we show that good low-rank approximations can be directly obtained from the Lanczos bidiagonalization process applied to the given matrix without computing any SVD. We ...

There has been considerable work on matrix approximation problems in the space of matrices with Euclidean and unitarily invariant norms. We initiate the study of approximation problems in the space P of all n×n positive definite matrices with the Riemannian metric δ2. Our main theorem reduces the approximation problem in P to an approximation problem in the space of Hermitian matrices and then ...

When data is sampled from an unknown subspace, principal component analysis (PCA) provides an effective way to estimate the subspace and hence reduce the dimension of the data. At the heart of PCA is the EckartYoung-Mirsky theorem, which characterizes the best rank k approximation of a matrix. In this paper, we prove a generalization of the Eckart-Young-Mirsky theorem under all unitarily invari...

The aim of this paper is to gain more insight into vector and matrix medians and to investigate algorithms to compute them. We prove relations between vector and matrix means and medians, particularly regarding the classical structure tensor. Moreover, we examine matrix medians corresponding to different unitarily invariant matrix norms for the case of symmetric 2 × 2 matrices which frequently ...

In graph theory a partition of the vertex set of a graph is called equitable if for all pairs of cells all vertices in one cell have an equal number of neighbours in the other cell. Considering the implications for the adjacency matrix one may generalize that concept as a block partition of a complex square matrix s.t. each block has constant row sum. It is well known that replacing each block ...