Aconjugate decomposition oftheEuclidean space (matrices/convex cones/positive definiteness/duality/optimizatio n)
S.-P. Han · 1983
ABSTRACT Given a closed convex cone Kin the n-dimen-sional real Euclidean spaceR'andan nxn real matrixAthat ispositive definite onK, weshowthateach vector inRncanbede- composed into a component thatlies inKand another thatlies in the conjugatecone induced byAandsuchthatthetwo vectors are conjugate toeachotherwithrespecttoA+ AT. Asa consequenceofthis decomposition we establish the following characterizationof positive definitematrices: AnnXnreal matrix Ais positive def-inite if and only if it is positive definite on some closed convex coneKinR'and(A + AT)-1 exists andis positive semidefinite onthe polar cone K0. If Kis a subspace of Rn, then K0 is its or- thogonalcomplement KV. Other applications include local dualityresultsfornonlinearprograms andother characterizations ofpos-itive definite and semidefinite matrices.LetAbeann X nreal matrixandKbea closed convexconeinthe n-dimensional real Euclidean spaceR'. Avectorain RIn issaid to have a conjugate decomposition with respect to AandKif there exists anxin Kandayin the