the connection between matrix and convex cone

Question

I'm trying to understand the connection between convex cone and matarix. according to Boyd as you can see in the pic: X is a p.s.d matrix but how this matrix represent a convex cone? and why the matrix is looking like this the the value 'y' appears twice? what am I missing from here?

Another thins is what should I do when the matrix is 3x3 or 4x4? how do I need to build the matrix then?

thnx in advanced

Answer 1

First, let’s look at the definition of a cone:

A subset $C$ of a vector space $V$ is a cone iff for all $x\in{C}$ and scalars $\alpha\in\mathbb{R}$ with $\alpha\geqslant0$, the vector $\alpha{x}\in{C}$.

So we are interested in the set $\mathbb{S}^n$ of positive semidefinite $n\times{n}$ matrices. All we need to do is check the definition above—i.e. check that for any $M\in\mathbb{S}^n$ and $\alpha\geqslant0$, the matrix $\alpha{M}$ is positive semidefinite (this isn’t so hard).

In 3 or 4 dimensions, you won’t be able to visualize this cone—it’s not necessarily helpful to consider the visualization. The use is in the definition above.

Also, the value $y$ appears twice because Boyd is talking about symmetric matrices.