
Schur complement proof
Trying to prove that if $\displaystyle A$ > $\displaystyle 0$, then $\displaystyle $M$$ is positive semidefinite if and only if $\displaystyle $C=B^*$$ and $\displaystyle $DCA^{1}B \ge 0$$
I have written
$\displaystyle M=\begin{pmatrix} A&B\\B^*&D \end{pmatrix}= \begin{pmatrix} 1&BD^{1}\\0&1 \end{pmatrix}\begin{pmatrix} ABD^{1}B^*&0\\0&D \end{pmatrix}\begin{pmatrix} 1&BD^{1}\\0&1 \end{pmatrix}^* = QPQ^*$
but this isn't quite right...I think the last matrix needs to be
$\displaystyle \begin{pmatrix} 1&0\\B^*D^{1}&1 \end{pmatrix} $
but then I can't write it as $\displaystyle QPQ^*, $ can I? Or are they the same?