So long story short. I have a matrix $A \in S^2_+$, that is, a symmetric, positive semi-definite 2x2 matrix. Here it is: $A = \begin{bmatrix} x & y \\y & z \end{bmatrix}$. Here is what it 'looks like' apparently.
I have two problems:
- The first one is that I would like to find out the constraints of the $x,y$, and $z$. For example, that $x \geq0$, $z \geq0$, and the same for $y$. I do not know how to find this. I have tried multiple things like using the characteristic polynomial etc, but no dice. I know that the eigenvalues must be $\geq0$ etc, but...
- Once I know what the x,y and z are, I think I will be able to interpret this diagram.
That is all, I appreciate any help. Thanks!
For 2x2, it's pretty easy: not only must the diagonal elements be positive, but the determinant must be, too. So the third constraint is $xz-y^2 \ge 0$, or $y^2 \le xz$. Coupled with the nonnegativity constraints on $x,z$, you can also write this as $|y| \le \sqrt{xz}$.