For some context, I see that when people try to determine the positive definiteness of a matrix, the Sylvester criterion can be of great help since the determinant is easy to calculate. For the case of block partition matrix, people seems to love the Schur complement.
In convex optimization, I have run into a situation where the set of inequality constraints $$ {x_1} \ge 0, \qquad {x_2} \ge 0, \qquad 1 - {x_1}{x_2} \le 0 $$ has been reformulated into a single linear matrix inequality (LMI) constraint.
$$\left[ {\begin{array}{*{20}{c}} {{x_1}}&1\\ 1&{{x_2}} \end{array}} \right]\succcurlyeq0$$
This gives me a feeling that the Sylvester crtiterion and the Schur complement are some how related.
So is there a relationship between Sylvester criterion and Schur complement for positive semidefiniteness of a matrix?
If this relationship exist, can we use this to reformulate non-convex constraints as LMI?