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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?

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    $\begingroup$ It may be impossible. If $p(x_0) = 0$, matrix $X$ isn't invertible, but $S^{-1}XS$ is. $\endgroup$
    – user207868
    Commented Apr 20, 2015 at 15:39

1 Answer 1

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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.

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  • $\begingroup$ In the case $p(x)$=-1 how would we find $S$? $\endgroup$
    – user119264
    Commented Apr 21, 2015 at 12:35
  • $\begingroup$ $S^{-1}=\left[\begin{smallmatrix}-1&1\\1&1\end{smallmatrix}\right]$; the columns are the eigenvectors of $X$. $\endgroup$
    – vadim123
    Commented Apr 21, 2015 at 13:51

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