schur complement of a block matrix I want to apply the Schur complement to one element of a block matrix. What I do not know is how to organize the resulting values inside the new matrix.
E.g.
Given the following block matrix with appropriate dimensions:
\begin{equation}
 \begin{bmatrix}
  A-BD^{-1}B^T & E \\
  E^T & C
 \end{bmatrix}\succ 0
\end{equation}
with $C=C^T$,$A=A^T$, $D=D^T$.
Applying Schur complement with respect to $A-BD^{-1}B^T$ yields to:
\begin{equation}
\begin{bmatrix}
 A & B & E \\
 B^T & D & 0\\
 E^T & 0 & C
\end{bmatrix}\succ 0
\end{equation}
or to\begin{equation}
 \begin{bmatrix}
  A & E & B \\
  E^T & C & 0\\
  B^T & 0 & D
 \end{bmatrix}\succ 0
\end{equation}
Which one of the two is correct? How to determine the order of the resulting matrix?
Thanks in advance!

The two $3 \times 3$ block matrices are similar via a block similarity, so one is positive definite if and only if the other is as well.  In particular:
$$
\pmatrix{I&0&0\\0&0&I\\0&I&0}
\pmatrix{A & B & E \\
 B^T & D & 0\\
 E^T & 0 & C}
\pmatrix{I&0&0\\0&0&I\\0&I&0}^T = 
\pmatrix{A & E & B \\
  E^T & C & 0\\
  B^T & 0 & D}
$$
To find the size of the resulting matrices, it suffices to know the sizes of $A,C,D$.
