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 $\alpha X+\beta Y\in P$