We prove explicitly that to every discrete, semibounded Hamiltonian with constant degeneracy and with finite sum of the squares of the reciprocal of its eigenvalues and whose eigenvectors span the entire Hilbert space there exists a characteristic self-adjoint time operator which is canonically conjugate to the Hamiltonian in a dense subspace of the Hilbert space. Moreover, we show that each ch...

State-of-the-art methods for learning unions of subspaces from a collection of data leverage sparsity to form representations of each vector in the dataset with respect to the remaining vectors in the dataset. The resulting sparse representations can be used to form a subspace affinity matrix to cluster the data into their respective subspaces. While sparsity-driven methods for subspace cluster...

Subspace clustering is an emerging task that aims at detecting clusters in entrenched in subspaces. Recent approaches fail to reduce results to relevant subspace clusters. Their results are typically highly redundant and lack the fact of considering the critical problem, “the density divergence problem,” in discovering the clusters, where they utilize an absolute density value as the density th...

Let T be a continuous linear operator on a Hausdorff topological vector space X over the field C. We show that if T is N -supercyclic, i.e., if X has an N dimensional subspace whose orbit under T is dense in X , then T ∗ has at most N eigenvalues (counting geometric multiplicity). We then show that N -supercyclicity cannot occur nontrivially in the finite dimensional setting: the orbit of an N ...

We compare two approaches to compute a fraction of the spectrum of dense symmetric definite generalized eigenproblems: one is based on the reduction to tridiagonal form, and the other on the Krylov-subspace iteration. Two large-scale applications, arising in molecular dynamics and material science, are employed to investigate the contributions of the application, architecture, and parallelism o...

We present an overview of the banded Invariant Subspace Decomposition Algorithm for symmetric matrices and describe a parallel implementation of this algorithm. The algorithm described here is a promising variant of the Invariant Subspace Decomposition Algorithm for dense symmetric matrices (SYISDA) that retains the property of using scalable primitives, while requiring signiicantly less overal...

Given the set of matrix pairs M ⊂ Mm,n(C) × Mn(C) keeping a subspace S ⊂ C invariant, we obtain a miniversal deformation of a pair belonging to an open dense subset of M. It generalizes the known results when S is a supplementary subspace of the unobservable one. Keywords-Conditioned invariant subspaces, Miniversal deformation, Stratified manifold, Vertical pairs of matrices.