For Given Matrix Q , I need help in construting new matrix P in order to QP will be positive definite. The Details and examples are as follows:
Let $Q=\begin{pmatrix} rI & A^\mathrm{T} \\ 0 & sI \\ \end{pmatrix}$, $P=\begin{pmatrix} B & C \\ D & E \\ \end{pmatrix}$, So that $QP$ is positive definite. Please give as more as possible forms of matrix $P$ and the conditions for positive definiteness of $QP$. ($r,s \in R$ are constants, $I$ is identity matrix.)
For example, \begin{eqnarray*} % \nonumber to remove numbering (before each equation) P_1 &=& \begin{pmatrix} I & \frac{1}{r}A^\mathrm{T} \\ 0 & I \\ \end{pmatrix}, ( rs \geq \|A^TA\| ), \\ P_2 &=& \begin{pmatrix} I & 0 \\ \frac{-1}{s}A & I \\ \end{pmatrix} ,(rs \geq \|A^TA\|), \\ P_3 &=& (1-\tau)P_1 + \tau P_2,( rs \geq \frac{3}{4}\|A^TA\|, \tau \in [0,1]), \\ P_4 &=& \begin{pmatrix} I & \frac{1}{2r}A^\mathrm{T} \\ \frac{-1}{2s}A & I-\frac{AA^T}{2rs} \\ \end{pmatrix}, ( rs \geq \frac{1}{4}\|A^TA\| ). \end{eqnarray*}