On matrix tridiagonalization

Question

Can any matrix $X \in \mathbb R^{n \times n}$ be decomposed into a tridiagonal matrix, i.e.,

$$X = P^{-1}DP$$

where $P \in \mbox{SO}(n)$ and $D$ is tridiagonal?

Answer 1

B. Sturmfels has proved the following result in the complex case:

For every $n\ge6$, there exists a complex $n\times n$ matrix that is not unitarily similar to a tridiagonal matrix.

A proof, due to J.L. Noakes, was given in W.E. Longstaff, On Tridiagonalization of Matrices, Linear and Its Applications, 109:153-163, 1988. Apparently, the dimension argument employed in this proof can be extended to the real case (and the answer then is negative when $n\ge5$).