Why does a positive definite matrix defines a convex cone?

I've been working on convex optimization and got stuck.

What exactly does a positive definite(p.d) matrix represent geometrically ? what kind of vector space it forms ?

If I have a p.d matrix which represent a convex cone (which I can't understand why), how do I prove the convexity for that matrix ? What's the input variable say X should be ?

Say if I have a plane, $$W^TX = B$$

at least I know I should put X into the equation, but for a p.d matrix...

it's just a matrix, why does that even represent a function ?

I am totally confused. Any hint helps a lot.

Ok. Let $P$ be the set of all positive definite matrix. Im gonna show that if $X,Y\in P$ and $\alpha,\beta>0$ then $\alpha X+\beta Y\in P$. Note \begin{eqnarray} x^{\top}(\alpha X+\beta Y)x &=& \alpha x^{\top} Xx+\beta x^{\top} Yx \nonumber \\ &>& 0\end{eqnarray}

Above, i have used algebraic properties of matrix product and the positive definitiness of $X$ and $Y$. With this you can conclude that $\alpha X+\beta Y\in P$

Let us take a simple positive definite matrix: $A = \bigl(\begin{smallmatrix} 1&0 \\ 0&1 \end{smallmatrix} \bigr)$

Let $\vec{x}=(x_1, x_2)$ be a vector such as $\vec{x}\in \mathcal{R}^2$, $\vec{x} \neq \vec{0}$

Now compute the quadratic form: $\vec{x}^\top A \vec{x} = x_1^2 + x_2^2$; which is definitely convex, since the square term is a convex function and the sum of two convex terms is convex. You can verify by plotting the above function