TITLE: How do I prove that $x^TAy = y^TAx$ if A is symmetric?
QUESTION [7 upvotes]: Ok this is for a HW but I'm not looking for a handout...just a hint to get me on the right track.
I have no idea where to begin proving this:
Show that if A is a symmetric matrix, then
$$x^TAy = y^TAx$$

REPLY [5 votes]: If $A$ is symmetric then we know that $A_{ij} = A_{ji}$.
If you understand that  $x^T A y$ = $\sum_i\sum_j x_iA_{ij}y_j$ , then swapping the indices of $A$ should directly lead you to the answer.