Rank property of a matrix including symmetric and persymmetric Hankel matrix

Question

I am investigating the property of the Hankel Matrix recently. Here, I have a question, what is the rank property of $[\mathbf{S}\ \mathbf{P}]$, where $\mathbf{S}$ is a symmetric hankel matrix and $\mathbf{P}$ is a persymmetric hankel matrix (both equal)?

P.S. I try to search the papers but all discussing Toeplitz plus Hankel matrix. I could not find any paper about this problem. I find that this matrix is full rank or rank one (all the elements are equal to a constant). Could anyone help me out? Thanks in advance!

Answer 1

Take $S=\begin{pmatrix}1&0&0\\0&0&0\\0&0&0\end{pmatrix}$ and $P=\begin{pmatrix}1&1&1\\1&1&1\\1&1&1\end{pmatrix}$; then the $3\times 6$ matrix $[SP]$ has rank $2$.