linear algebra questions about matrices Given a a matrix $X$ where 
$$
X=
  \begin{bmatrix}
    1 & 0  \\
    0 & p(x) 
  \end{bmatrix}
$$
where $p(x)$ is some polynomial of degree $3$ or 4 and different from $0$. I'm trying to find a matrix $S$ such that 
$$
S^{-1}XS=
  \begin{bmatrix}
    0 & -1  \\
    -1 & 0 
  \end{bmatrix}
$$
I'm not sure how to find $S$.. Any hints?

It is impossible to find what you seek.  The matrices $X$ and $S^{-1}XS$ are similar, and therefore have the same determinant.  The two determinants are $p(x)$ and $-1$, respectively.  Hence the only polynomial for which such an $S$ exists is the constant polynomial $p(x)=-1$; no polynomial of degree $3$ or $4$ can work.
