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

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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