Hello everyone. This is my first official post here but I have been lurking around for about a year now. 1. The problem statement, all variables and given/known data Prove that a matrix A is symmetric if and only if x*Ay = Ax*y for all x,y of R^n, where * denotes the dot product. 2. Relevant equations 3. The attempt at a solution So I was able to do prove the forward direction, as follows: Assume A is symmetric. Then x*Ay = x^T A y = x^T A^T y = (Ax)^T y = Ax * y as required. However, I am completely stumped for the other direction. I.e., assuming x*Ay = Ax*y and then showing A = A^T. Any suggestions?