If the quadratic form of $Q(x)=x^T Ax$ and $\langle x,y\rangle=\frac{1}{2}[Q(x+y)-Q(x)-Q(y)]$. Where $x, y$ are vectors in $\def\R{\Bbb R}\R^n$ and $A$ is a $n\times n$ matrix. Show that $\langle\, , \rangle$ is an inner product on $\R^n$ if and only if $Q(x)$ is positive definite.
If $\{v_1,...,v_k\}$ are the eigenvectors of $A$, $\langle v_i,v_i\rangle=\lambda_i\langle v_i,v_i\rangle$. So all eigenvalues must be positive. Not sure how to continue.
<,> is bilinear, you have to show it is definite positive if and only if $Q$ is definite positive you have $\langle x,x \rangle = Q(x)$. Thus $<x,x>>0 $ if and only if $Q(x)>0$.