Given the real square block matrix $ M = \begin{pmatrix} A & D^t & 0\\ D & c E & c F\\ 0 & H & J \end{pmatrix} $ how can I make it symmetric? Note that the dimension of $F$ is $n \times m$ and the dimension of $H$ is $m \times n$ so that I think I require $H = F^t$ to make it symmetric.
Wikipedia states that I can compute matrices $D$ and $S$ such that $M = DS$ where $D$ is a diagonal matrix and $S$ is a symmetric matrix, the question is how do I get those matrices. I think that the matrices would look like $ D = \begin{pmatrix} I & 0 & 0 \\ 0 & ? & 0 \\ 0 & 0 & ? \end{pmatrix} $ and $ S = \begin{pmatrix} A & D^t & 0 \\ D & ? & ? \\ 0 & ? & ? \end{pmatrix} $ but again I am not sure how to compute them. Any help is really appreciated since I am completely lost, a reference link with an example would be of great help.
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