I am trying to understand the following proof from the book "Linear Matrix Inequalities in System and Control Theory".
However I am struggling to understand why $S_{2}$ must equal zero. Why isn't it enough for $S_{2}$ to simply be Positive Semi-Definite?
$S_2$ is not necessarily a square matrix. You cannot speak of its definiteness when it is not square.
Anyway, for any real number $t$ and any pair of vectors $x$ and $z$. Therefore, by replacing $z$ by $-z$ if necessary, we have $x^TS_2z<0$. Hence $$ \pmatrix{x^T&0&tz^T}\pmatrix{Q&S_1&S_2\\ S_1^T&\Sigma&0\\ S_2^T&0&0}\pmatrix{x\\ 0\\ tz}=x^Tx+2tx^TS_2z<0 $$ when $t>0$ is sufficiently large.
Alternatively, if $S_2\ne0$, then the block matrix contains a $2\times2$ principal submatrix of the form $\pmatrix{q&s_2\\ s_2&0}$ where $q$ is picked from $Q$ and $s_2$ is picked from $S_2$. Surely $\pmatrix{q&s_2\\ s_2&0}$ is indefinite. Therefore the block matrix is indefinite as well.