Let $n \times n$ matrix $H_n$ be anti-tridiagonal. For example, when $n = 4$,
$$H_4 = \begin{bmatrix} 0 & 0 & a & b\\ 0 & a & b & c\\ a & b & c & 0\\ b &c & 0 & 0\end{bmatrix}$$
If $J_n$ is the $n \times n$ backward identity and $H_n$ is persymmetric, I know that $H = J_n T_n = T_n J_n$, where $T_n$ is tridiagonal.
If $H$ is not persymmetric, can I matrix $H$ similarity to persymmetric anti-tridiagonal?