# How can i prove the following function is positive?

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.

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