I want to devise a retraction map for a particular optimization problem, which needs to be locally an affine map. The problem is reduced to the case that, does there exist a linear map $T$ with respect to $v$, where $T: Sym(n) \times \mathbb{R}^{k \times n} \to \mathbb{R}^{k \times n}$ on the space of symmetric matrices such that
$$T(v, x)^Tx + x^TT(v, x) = v,$$
for any symmetric matrix $v \in Sym(n) $, any positive matrix $x \in \mathbb{R}^{k \times n}$? I do not think that there exists such a map $T$, but have no clue on how to show it exists or it does not exist. Any hints will be much appreciated.