1,342 research outputs found
Gershgorin disks for multiple eigenvalues of non-negative matrices
Gershgorin's famous circle theorem states that all eigenvalues of a square
matrix lie in disks (called Gershgorin disks) around the diagonal elements.
Here we show that if the matrix entries are non-negative and an eigenvalue has
geometric multiplicity at least two, then this eigenvalue lies in a smaller
disk. The proof uses geometric rearrangement inequalities on sums of higher
dimensional real vectors which is another new result of this paper
Spectral theorem for the Lindblad equation for quadratic open fermionic systems
The spectral theorem is proven for the quantum dynamics of quadratic open
systems of n fermions described by the Lindblad equation. Invariant eigenspaces
of the many-body Liouvillean dynamics and their largest Jordan blocks are
explicitly constructed for all eigenvalues. For eigenvalue zero we describe an
algebraic procedure for constructing (possibly higher dimensional) spaces of
(degenerate) non-equilibrium steady states.Comment: 19 pages, no figure
Local normal forms for c-projectively equivalent metrics and proof of the Yano-Obata conjecture in arbitrary signature. Proof of the projective Lichnerowicz conjecture for Lorentzian metrics
Two K\"ahler metrics on a complex manifold are called c-projectively
equivalent if their -planar curves coincide. These curves are defined by the
property that the acceleration is complex proportional to the velocity. We give
an explicit local description of all pairs of c-projectively equivalent
K\"ahler metrics of arbitrary signature and use this description to prove the
classical Yano-Obata conjecture: we show that on a closed connected K\"ahler
manifold of arbitrary signature, any c-projective vector field is an affine
vector field unless the manifold is with (a multiple of) the
Fubini-Study metric. As a by-product, we prove the projective Lichnerowicz
conjecture for metrics of Lorentzian signature: we show that on a closed
connected Lorentzian manifold, any projective vector field is an affine vector
field.Comment: comments are welcom
Tridiagonal realization of the anti-symmetric Gaussian -ensemble
The Householder reduction of a member of the anti-symmetric Gaussian unitary
ensemble gives an anti-symmetric tridiagonal matrix with all independent
elements. The random variables permit the introduction of a positive parameter
, and the eigenvalue probability density function of the corresponding
random matrices can be computed explicitly, as can the distribution of
, the first components of the eigenvectors. Three proofs are given.
One involves an inductive construction based on bordering of a family of random
matrices which are shown to have the same distributions as the anti-symmetric
tridiagonal matrices. This proof uses the Dixon-Anderson integral from Selberg
integral theory. A second proof involves the explicit computation of the
Jacobian for the change of variables between real anti-symmetric tridiagonal
matrices, its eigenvalues and . The third proof maps matrices from the
anti-symmetric Gaussian -ensemble to those realizing particular examples
of the Laguerre -ensemble. In addition to these proofs, we note some
simple properties of the shooting eigenvector and associated Pr\"ufer phases of
the random matrices.Comment: 22 pages; replaced with a new version containing orthogonal
transformation proof for both cases (Method III
An Accelerated Conjugate Gradient Algorithm to Compute Low-Lying Eigenvalues --- a Study for the Dirac Operator in SU(2) Lattice QCD
The low-lying eigenvalues of a (sparse) hermitian matrix can be computed with
controlled numerical errors by a conjugate gradient (CG) method. This CG
algorithm is accelerated by alternating it with exact diagonalisations in the
subspace spanned by the numerically computed eigenvectors. We study this
combined algorithm in case of the Dirac operator with (dynamical) Wilson
fermions in four-dimensional \SUtwo gauge fields. The algorithm is
numerically very stable and can be parallelized in an efficient way. On
lattices of sizes an acceleration of the pure CG method by a factor
of~ is found.Comment: 25 pages, uuencoded tar-compressed .ps-fil
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