I meet a difficult problem recently. The problem is to find the rank of a special matrix: $$X = \begin{bmatrix} S & P \\ P & JSJ\end{bmatrix}\in \mathbb{R}^{2m\times2m} ,$$ where $S$ is symmetrix matrix (actually is a hankel matrix) and $P$ is Persymmetric matrix and $J$ is Exchange matrix.
I want to find out whether the case $detA = 0$ exists. One possible solution is to decompose the $X$ into multiplication of some block matrix. But hwo to do it? Could anyone provide some hints? Thanks in advance!