Many inequality relations between real vector quantities can be succinctly expressed as “weak (sub)majorization” relations using the symbol ≺w. We explain these ideas and apply them in several areas: angles between subspaces, Ritz values, and graph Laplacian spectra, which we show are all surprisingly related. Let Θ(X ,Y) be the vector of principal angles in nondecreasing order between subspace...

We study the Majorization arrow in a big class of quantum adiabatic algorithms. In a quantum adiabatic algorithm, the ground state of the Hamiltonian is a guide state around which the actual state evolves. We prove that for any algorithm of this class, step-by-step majorization of the guide state holds perfectly. We also show that step-by-step majorization of the actual state appears if the run...

Majorization is a powerful, easy-to-use and exible tool which arises frequently in quantum mechanics as a consequence of fundamental connections between unitarity and the majorization relation. Entanglement theory does not escape from its in uence. Thus the interconversion of bipartite pure states by means of local manipulations turns out to be ruled to a great extend by majorization relations....

DIAGONAL ELEMENTS, EIGENVALUES, AND SINGULAR VALUES DRAFT AS OF April 30, 2013 SHENG-JHIH WU AND MOODY T. CHU Abstract. Diagonal entries and eigenvalues of a Hermitian matrix, diagonal entries and singular values of a general matrix, and eigenvalues and singular values of a general matrix satisfy necessarily some majorization relationships which turn out also to be sufficient conditions. The in...

We describe an algorithm for radial layout of undirected graphs, in which nodes are constrained to the circumferences of a set of concentric circles around the origin. Such constraints frequently occur in the layout of social or policy networks, when structural centrality is mapped to geometric centrality, or when the primary intention of the layout is the display of the vicinity of a distingui...

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 9 July 2021Accepted: 21 September 2021Published online: 13 January 2022Keywordsinertial block coordinate method, majorization minimization, Bregman surrogate function, acceleration by extrapolation, orthogonal nonnegative matrix factorizationAMS Subject Headings90C26, 49M...

In multivariate statistical analysis. orthogonally invariant sets of real positive definite pxp matrices occur as acceptance regions for tests of invariant hypotheses concerning the covariance matrix [ of a multivariate normal distribution. Equivalently. orthogonally invariant acceptance regions can be expressed in terms of the eigenvalues I, (S). .... lp(S) of a random Wishart matrix S Wp(n. [...

In this paper, we aim at finding a nearest correlation matrix to a given symmetric matrix, measured by the componentwise weighted Frobenius norm, with a prescribed rank and bound constraints on its correlations. This is in general a non-convex and difficult problem due to the presence of the rank constraint. To deal with this difficulty, we first consider a penalized version of this problem and...