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I have a little question about similarity of matrices.

If $A,B\in\mathrm{Mat}_n(\mathbf{R})$, and we want to show $A$ and $B$ are similar over ''$\mathbf{R}$''.

I need to show they have same ''rational canonical form'' or ''Jordan form''?

(Because, $\mathbf{R}$ is not algebraic closed, so does that means we need to say they have same rational canonical form? Is it sufficient to just prove they have same Jordan form?)

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    Similarity over $\mathbb{C}$ is equivalent as similarity over $\mathbb{R}$. See for example https://math.stackexchange.com/questions/57242/similar-matrices-and-field-extensions?noredirect=1&lq=1 Therefore, it suffices to prove they have the same Jordan form. – Zoudelong Apr 30 '25 at 04:19

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