# Concavity of certain maps on positive definite matrices and applications to Hadamard products

@article{And1979ConcavityOC, title={Concavity of certain maps on positive definite matrices and applications to Hadamard products}, author={T. And{\^o}}, journal={Linear Algebra and its Applications}, year={1979}, volume={26}, pages={203-241} }

Abstract If f is a positive function on (0, ∞) which is monotone of order n for every n in the sense of Lowner and if Φ1 and Φ2 are concave maps among positive definite matrices, then the following map involving tensor products: (A,B)↦f[Φ 1 (A) −1 ⊗Φ 2 (B)]·(Φ 1 (A)⊗I) is proved to be concave. If Φ1 is affine, it is proved without use of positivity that the map (A,B)↦f[Φ 1 (A)⊗Φ 2 (B) −1 ]·(Φ 1 (A)⊗I) is convex. These yield the concavity of the map (A,B)↦A 1−p ⊗B p (0 (A,B)↦A 1+p ⊗B −p (0 (A,B… Expand

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