Let $X$ and $Y$ be $n \times m$ matrices. The matrix $$ A = X^TY + Y^TX $$ will be a $m \times m$ square, symmetric matrix. Is it possible to say: i) when is $A$ is positive definite? and when it is ii) letting $A = Z^TZ$, can we express $Z$ in terms of $X$ and $Y$? I assume the two questions are related since the second would imply the first but I am not sure how to proceed.
Edit: the question came about as I want to show that $A$ in the sum $$ A = X^TSY + Y^TSX $$ where $S$ is a $n \times n$ symmetric positive definite matrix, is positive definite. The presence of $S$ may make the problem simpler.