Let $A$ a symmetric matrix, $x\in X$ where $\left(X,\left<\cdot \right>\right)$ is a inner product space, $\beta\in\mathbb R\backslash \{0\}$ and $v=\frac{Ax}{\beta}$. My question is probably obvious, but I don't understand where the following equality comes from:
$$\|Ax\|^2=\frac{1}{4}\left(2\left<A(\beta x), v\right>+2\left<Av,\beta x\right>\right)$$
Plugging in the definition for $v$, we have by bilinearity of the inner product
$$\frac{1}{2} (\langle A \beta x, A \beta^{-1} x \rangle + \langle A^2 b^{-1} x, \beta x \rangle) = \frac{1}{2} (\langle Ax, Ax \rangle + \langle A^2x, x \rangle) $$
and since $A$ is symmetric, $\langle A^2x, x \rangle = \langle Ax, Ax \rangle$, so we are left with
$$\langle Ax, Ax \rangle = \| Ax \|^2.$$