Let $A\in GL_n(\mathbb R)$ and $Q:\ \mathbb R^n\longrightarrow\mathbb R$ is a quadratic form such that $\big[Q\big]_{\mathcal B}=A^\top A$ for some basis $\mathcal B$ of $\mathbb R^n$. Then $Q$ is positive definite.
My solution : for all $u\in\mathbb R^n$, we have \begin{align} Q(u)&=\big[u\big]_{\mathcal B}^\top\big[Q\big]_{\mathcal B}\big[u\big]_{\mathcal B} \\ &=\big[u\big]_{\mathcal B}^\top A^\top A\big[u\big]_{\mathcal B} \\ &=\big(A[u]_{\mathcal B}\big)^\top A\big[u\big]_{\mathcal B} \\ &=\big|v\big|^2\,\geq\,0 \end{align} where $[v]=A[u]_{\mathcal B}$.
Suppose $Q(u)=0$, then $|v|=0$, which implies $v=0$. Therefore $Q$ is positive definite.
Is my proof correct ? What do you think ?