Positive definiteness of a bilinear form implies symmetry?

Question

In the Wikipeda article about positive definite bilinear forms, there is the line

It turns out that the matrix $M$ is positive definite if and only if it is symmetric and its quadratic form is a strictly convex function.

What does it mean a strictly convex function? Is it strictly convex on every convex subset?

Answer 1

If $M$ is positive definite then Sylvester theorem says that $M$ has only positive eigenvalues and its quadratic form is the following:

$$q(x_1,..,x_n)=\lambda_1x_1^2+...+\lambda_nx_m^2$$ where $\lambda_i$ are the eigenvalues.

So the function is strictly convex since the Jacobian matrix for the function $q:\mathbb{R}^n\longrightarrow \mathbb{R}$ is $2M$ which is positive definite by assumption.