On positive definite preserving linear transformations of rank $r$ on real symmetric matrices

Details On positive definite preserving linear transformations of rank $r$ on real symmetric matrices On positive definite preserving linear transformations of rank $r$ on real symmetric matrices

2010-11-16

arxiv.org/abs/1011.3739v2

Mathematics

Operator Algebras

5 pages

Scientific paper

We study on what conditions on $B_k,$ \ a linear transformation of rank $r$ \label{form} T(A)=\sum_{k=1}^r\tr(AB_k)U_k where $U_k,\ k=1,2,..., r$ are linear independent and all positive definite; is positive definite preserving. We give some first results for this question. For the case of rank one and two, the necessary and sufficient conditions are given. We also give some sufficient conditions for the case of rank $r.$

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