SVD for low-rank approximaiton of non-symmetric matrix

Question

I have a square, real-valued, non-symmetric matrix $D$ that satisfies $D + D^T \geq 0$.

I want to use SVD to find a low-rank approximation of $D$.

Let's use $E$ to denominate the low-rank approximation of $D$, obtained via SVD.

Will $E$ also satisfy $E+E^T \geq 0$?