Positive definite preserving linear transformations on symmetric matrix spaces

Base on some simple facts of Hadamard product, characterizations of positive definite preserving linear transformations on real symmetric matrix spaces with an additional assumption “$\text{ra}T(E_{ii})=1,i=1,2,\mathrm{.}\mathrm{.}\mathrm{.},n$” or “$T(A)>0\to A> 0$”, were given.