Why is $(AX)\cdot Y=X\cdot (A^{T}Y)$?
$$(AX)\cdot Y = \lambda X\cdot Y$$ $$Y\cdot (A^TX) = Y\cdot X \lambda = \lambda X\cdot Y$$ $$\Rightarrow (AX)\cdot Y = X\cdot(A^T Y)$$
Remember that in a symmetric matrix, $A = A^T$, and that the dot product of vectors is commutative.