Unsymmetric tridiagonal matrices $T_3$ can easily be symmetrized via a (diagonal) similarity transformation $D=\text{diag}(d_1, \dots, d_n)$ (i.e. see Wikipedia) $$ J_3=D^{-1} T_3 D \,. $$
Is there a similarity transformation $P$ for pentadiagonal matrices $T_5$ such that the transformed matrix $$ J_5 = P^{-1} T_5 P $$ is symmetric? Possibly using a lower bidiagonal matrix as $P$?
PS: This post is a double. I also posted this question at mathoverflow.