矩阵的Real canonical form

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矩阵$A$转换为实规范形\[A=\begin{pmatrix}
0 & 0 & 0&-1\\
1 & 0 & 0& 0\\
0 & 1 & 0 & 0\\
0& 0& 1& 0
\end{pmatrix}\]
$\chi_A(\lambda)=\lambda^4+1\implies\lambda=\frac{\pm1\pm i}{\sqrt2}$
$$\therefore A\sim\begin{pmatrix}
\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} & 0 & 0 \\
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & 0 & 0 \\
0 & 0 & -\frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\
0 & 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\
\end{pmatrix}$$

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$A$转换为实规范形(Real canonical form)与$A$转换为有理规范形(Frobenius normal form)相同吗？