Consider a matrix, $$M = x y^{\top}.$$
Say, $$M_{\text{sym.}} = xy^{\top}+yx^{\top}$$ is the symmetric part of $M$.
How can one represent $M_{\text{sym.}}$ in the form $$M_{\text{sym.}} = zz^{\top},$$ where $z$ is a function of vectors $x$ and $y$: $$z = f(x,y)?$$
P.S.: Assume all vectors have appropriate dimensions.