752 research outputs found
Tridiagonal and Pentadiagonal Doubly Stochastic Matrices
We provide a decomposition that is sufficient in showing when a symmetric tridiagonal matrix A is completely positive and provide examples including how one can change the initial conditions or deal with block matrices, which expands the range of matrices to which our decomposition can be applied. Our decomposition leads us to a number of related results, allowing us to prove that for tridiagonal doubly stochastic matrices, positive semidefiniteness is equivalent to complete positivity (rather than merely being implied by complete positivity). We then consider symmetric pentadiagonal matrices, proving some analogous results, and providing two different decompositions sufficient for complete positivity, again illustrated by a number of examples
Tridiagonal and Pentadiagonal Doubly Stochastic Matrices
We provide a decomposition that is sufficient in showing when a symmetric
tridiagonal matrix is completely positive and provide examples including
how one can change the initial conditions or deal with block matrices, which
expands the range of matrices to which our decomposition can be applied. Our
decomposition leads us to a number of related results, allowing us to prove
that for tridiagonal doubly stochastic matrices, positive semidefiniteness is
equivalent to complete positivity (rather than merely being implied by complete
positivity). We then consider symmetric pentadiagonal matrices, proving some
analogous results, and providing two different decompositions sufficient for
complete positivity, again illustrated by a number of examples.Comment: 15 page
An atlas for tridiagonal isospectral manifolds
Let be the compact manifold of real symmetric tridiagonal
matrices conjugate to a given diagonal matrix with simple spectrum.
We introduce {\it bidiagonal coordinates}, charts defined on open dense domains
forming an explicit atlas for . In contrast to the standard
inverse variables, consisting of eigenvalues and norming constants, every
matrix in now lies in the interior of some chart domain. We
provide examples of the convenience of these new coordinates for the study of
asymptotics of isospectral dynamics, both for continuous and discrete time.Comment: Fixed typos; 16 pages, 3 figure
Fundamental length in quantum theories with PT-symmetric Hamiltonians II: The case of quantum graphs
Manifestly non-Hermitian quantum graphs with real spectra are introduced and
shown tractable as a new class of phenomenological models with several
appealing descriptive properties. For illustrative purposes, just equilateral
star-graphs are considered here in detail, with non-Hermiticities introduced by
interactions attached to the vertices. The facilitated feasibility of the
analysis of their spectra is achieved via their systematic approximative
Runge-Kutta-inspired reduction to star-shaped discrete lattices. The resulting
bound-state spectra are found real in a discretization-independent interval of
couplings. This conclusion is reinterpreted as the existence of a hidden
Hermiticity of our models, i.e., as the standard and manifest Hermiticity of
the underlying Hamiltonian in one of less usual, {\em ad hoc} representations
of the Hilbert space of states in which the inner product is local
(at ) or increasingly nonlocal (at ). Explicit examples of
these (of course, Hamiltonian-dependent) hermitizing inner products are offered
in closed form. In this way each initial quantum graph is assigned a menu of
optional, non-equivalent standard probabilistic interpretations exhibiting a
controlled, tunable nonlocality.Comment: 33 pp., 6 figure
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