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Theory of cones

Linear Algebra and its Applications
Publication Date
DOI: 10.1016/0024-3795(81)90310-4
  • Mathematics


Abstract This survey deals with the aspects of archimedian partially ordered finite-dimensional real vector spaces and order preserving linear maps which do not involve spectral theory. The first section sketches some of the background of entrywise nonnegative matrices and of systems of inequalities which motivate much of the current investigations. The study of inequalities resulted in the definition of a polyhedral cone K and its face lattice F( K). In Section II.A the face lattice of a not necessarily polyhedral cone K in a vector space V is investigated. In particular the interplay between the lattice properties of F ( K) and geometric properties of K is emphasized. Section II.B turns to the cones Π( K) in the space of linear maps on V. Recall that Π( K) is the cone of all order preserving linear maps. Of particular interest are the algebraic structure of Π( K) as a semiring and the nature of the group Aut( K) of nonsingular elements AϵΠ( K) for which A -1ϵΠ( K) as well. In a short final section the cone P n of n× n positive semidefinite matrices is discussed. A characterization of the set of completely positive linear maps is stated. The proofs will appear in a forthcoming paper.

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