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This should be a comment, but I can not yet post comments. You got Schur's complement lemma wrong, the matrix $$\left(\begin{array}{cc} 1 & 1 \newline 1 & -1 \end{array}\right)$$ satisfies you conditions, $A=1\ge 0$ and $A-BC^{-1}B^T=1-(-1)\ge 0$, but it is clearly not positive semi-definite. Replace the first condition by $C\ge C > 0$ (see also http://en.wikipedia.org/wiki/Schur_complement). Then Chris Godsil's comment shows that the answer to your first question is yes. |
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This should be a comment, but I can not yet post comments. You got Schur's complement lemma wrong, the matrix $$\left(\begin{array}{cc} 1 & 1 \newline 1 & -1 \end{array}\right)$$ satisfies you conditions, $A=1\ge 0$ and $A-BC^{-1}B^T=1-(-1)\ge 0$, but it is clearly not positive semi-definite. Replace the first condition by $C\ge 0$ (see also http://en.wikipedia.org/wiki/Schur_complement). Then Chris Godsil's comment shows that the answer to your first question is yes. |
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