1,038 research outputs found
PPT from spectra
In this contribution we solve the following problem. Let H_{nm} be a Hilbert
space of dimension nm, and let A be a positive semidefinite self-adjoint linear
operator on H_{nm}. Under which conditions on the spectrum has A a positive
partial transpose (is PPT) with respect to any partition H_n \otimes H_m of the
space H_{nm} as a tensor product of an n-dimensional and an m-dimensional
Hilbert space? We show that the necessary and sufficient conditions can be
expressed as a set of linear matrix inequalities on the eigenvalues of A.Comment: 6 pages, no figure
Minimal zeros of copositive matrices
Let be an element of the copositive cone . A zero of
is a nonzero nonnegative vector such that . The support of is
the index set \mbox{supp}u \subset \{1,\dots,n\} corresponding to the
positive entries of . A zero of is called minimal if there does not
exist another zero of such that its support \mbox{supp}v is a strict
subset of \mbox{supp}u. We investigate the properties of minimal zeros of
copositive matrices and their supports. Special attention is devoted to
copositive matrices which are irreducible with respect to the cone of
positive semi-definite matrices, i.e., matrices which cannot be written as a
sum of a copositive and a nonzero positive semi-definite matrix. We give a
necessary and sufficient condition for irreducibility of a matrix with
respect to in terms of its minimal zeros. A similar condition is given
for the irreducibility with respect to the cone of entry-wise
nonnegative matrices. For matrices which are irreducible with respect
to both and are extremal. For a list of candidate
combinations of supports of minimal zeros which an exceptional extremal matrix
can have is provided.Comment: Some conditions and proofs simplifie
Analytic formulas for complete hyperbolic affine spheres
We classify all regular three-dimensional convex cones which possess an
automorphism group of dimension at least two, and provide analytic expressions
for the complete hyperbolic affine spheres which are asymptotic to the
boundaries of these cones. The affine spheres are represented by explicit
hypersurface immersions into three-dimensional real space. The generic member
of the family of immersions is given by elliptic integrals.Comment: 16 page
Centro-affine hypersurface immersions with parallel cubic form
We consider non-degenerate centro-affine hypersurface immersions in R^n whose
cubic form is parallel with respect to the Levi-Civita connection of the affine
metric. There exists a bijective correspondence between homothetic families of
proper affine hyperspheres with center in the origin and with parallel cubic
form, and K\"ochers conic omega-domains, which are the maximal connected sets
consisting of invertible elements in a real semi-simple Jordan algebra. Every
level surface of the omega function in an omega-domain is an affine complete,
Euclidean complete proper affine hypersphere with parallel cubic form and with
center in the origin. On the other hand, every proper affine hypersphere with
parallel cubic form and with center in the origin can be represented as such a
level surface. We provide a complete classification of proper affine
hyperspheres with parallel cubic form based on the classification of
semi-simple real Jordan algebras. Centro-affine hypersurface immersions with
parallel cubic form are related to the wider class of real unital Jordan
algebras. Every such immersion can be extended to an affine complete one, whose
conic hull is the connected component of the unit element in the set of
invertible elements in a real unital Jordan algebra. Our approach can be used
to study also other classes of hypersurfaces with parallel cubic form.Comment: Fourth version, 35 pages. A missing case has been added to the
classificatio
Graph immersions with parallel cubic form
We consider non-degenerate graph immersions into affine space whose cubic form is parallel with respect to the Levi-Civita
connection of the affine metric. There exists a correspondence between such
graph immersions and pairs , where is an -dimensional real
Jordan algebra and is a non-degenerate trace form on . Every graph
immersion with parallel cubic form can be extended to an affine complete
symmetric space covering the maximal connected component of zero in the set of
quasi-regular elements in the algebra . It is an improper affine hypersphere
if and only if the corresponding Jordan algebra is nilpotent. In this case it
is an affine complete, Euclidean complete graph immersion, with a polynomial as
globally defining function. We classify all such hyperspheres up to dimension
5. As a special case we describe a connection between Cayley hypersurfaces and
polynomial quotient algebras. Our algebraic approach can be used to study also
other classes of hypersurfaces with parallel cubic form.Comment: some proofs have been simplified with respect to the first versio
Extremal copositive matrices with minimal zero supports of cardinality two
Let be an extremal copositive matrix with unit diagonal.
Then the minimal zeros of all have supports of cardinality two if and only
if the elements of are all from the set . Thus the extremal
copositive matrices with minimal zero supports of cardinality two are exactly
those matrices which can be obtained by diagonal scaling from the extremal
unit diagonal matrices characterized by Hoffman and Pereira in
1973.Comment: 4 page
Spectrahedral cones generated by rank 1 matrices
Let be the cone of positive semi-definite
matrices as a subset of the vector space of real symmetric
matrices. The intersection of with a linear subspace of is called a spectrahedral cone. We consider spectrahedral cones such
that every element of can be represented as a sum of rank 1 matrices in
. We shall call such spectrahedral cones rank one generated (ROG). We show
that ROG cones which are linearly isomorphic as convex cones are also
isomorphic as linear sections of the positive semi-definite matrix cone, which
is not the case for general spectrahedral cones. We give many examples of ROG
cones and show how to construct new ROG cones from given ones by different
procedures. We provide classifications of some subclasses of ROG cones, in
particular, we classify all ROG cones for matrix sizes not exceeding 4. Further
we prove some results on the structure of ROG cones. We also briefly consider
the case of complex or quaternionic matrices. ROG cones are in close relation
with the exactness of semi-definite relaxations of quadratically constrained
quadratic optimization problems or of relaxations approximating the cone of
nonnegative functions in squared functional systems.Comment: Version 2: section on complex and quaternionic case added, many
sections completely rewritte
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