Q:

Condition on the positivity of a quadratic form

We place ourself in $\mathbb{R}^{n}$.
Let's consider a positive definite matrix $M \in \mathcal{M}_{n} (\mathbb{R})$, $V$ and $E$ $\in \mathbb{R}^{n}$, and $\alpha > 0$.
We consider the application
$$\begin{aligned}
f : &\mathbb{R}^{n} \to \mathbb{R} \\ &X \mapsto \bigg[ M \, (V + X) \bigg] \cdot \bigg[ E + \alpha X \bigg]
\end{aligned}$$
I have the feeling that the following statement holds
$\forall X \;,\; f(X) \geq 0 \;\; \Longrightarrow \;\; E = \alpha V$
In the case $n=1$, one just needs to study the discriminant of this polynom of degree 2, which reads $\Delta = D^{2} (E - \alpha V)^{2}$, so that we find the expected condition.
Unfortunately, I am not at ease with quadratic forms for $n > 1$.
How would you handle the case of higher dimensions ? The idea would be to look at a kind of $\textit{discriminant}$ for higher dimensions...

A:

Hint: put $X = -V - \frac{1}{2\alpha}(E-\alpha V)$. Then $f(X)=-\frac{1}{4\alpha}(E-\alpha V)^TM(E-\alpha V)$.

