How can i prove the following function is positive?

Question

I have the following function.

$F =[x_1,x_2,...,x_n]_{1 \times n}*M_{n \times n}*([\dfrac{x_1}{|x_1|^{1/2}},\dfrac{x_2}{|x_2|^{1/2}}, ..., \dfrac{x_n}{|x_n|^{1/2}}]^T)_{n \times 1}$

Which $x \in \mathbb{R}$ and $M$ is a square matrix. In general, $F$ is not a positive function but when i choose $M$ as a positive definite matrix, $F$ is always positive (as i test it by MATLAB coding).

Now i need to prove $F$ is always positive (by choosing positive definite matrix $M$) mathematically, and i need your guidance.

Thanks in advance.

Answer 1

$$ \begin{pmatrix}1\\49\end{pmatrix}^\top \begin{pmatrix}197&-14\\-14&1\end{pmatrix} \begin{pmatrix}1\\7\end{pmatrix} = -244 $$