I require a positive definite matrix, $M$, with dimensions $2n\times2n$ of the form \begin{equation} M=\begin{pmatrix} \Sigma&P'\\ P&\Sigma \end{pmatrix} \end{equation} where $\Sigma$ is a $n\times{}n$ positive definite matrix. What are the constraints on $P$ that ensure that $M$ is positive definite?
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$\begingroup$ Is $P^{\prime}$ supposed to be the transpose of $P$? $\endgroup$– Geoff RobinsonApr 14, 2016 at 11:14
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$\begingroup$ Yes that is right $\endgroup$– M. SpenceApr 14, 2016 at 11:35
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4$\begingroup$ Using Schur complements we obtain the constraint: $\Sigma \ge P'\Sigma^{-1}P$ which is required for ensuring $M \ge 0$. $\endgroup$– SuvritApr 14, 2016 at 13:06
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