I know that my question might be trivial but I would appreciate your feedback.
I know that the Schur complement can be used to express a quadratic inequality as a positive semidefinite matrix and vice-versa. Why is that true? Is the Schur complement of a PSD matrix an equivalent of it? For example, in the image below
I know that positive semidefinite matrices, characterized by the non-negativity of their principal minors, i.e., for the linear matrix inequality (LMI)
$$\begin{bmatrix} x & y \\ y & z \end{bmatrix} \succeq 0 $$
this yields $x \ge 0$, $z \geq 0$ and $xz−y^2\ge0$.
Is $xz−y^2\ge0$ equivalent to the following LMI?
$$\begin{bmatrix} x & y \\ y & z \end{bmatrix} \succeq 0 $$
I'm really confused about the relationship between these inqualities and the LMI and the rule of Schur's complement in this? Please help!