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Assume we have a function $ f $ which maps matrix to its Jordan form (up to the order of the blocks). Would this function be linear?

Im asking because I want to know if Matrix similarity is linear, i.e given similar matrices $A,B$, and $C,D $, Do we know that $A+C $ is similar to $ B+D$?

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No. Consider the matrices $A=\begin{pmatrix} 1 & 1\\ 0 & 1\end{pmatrix}$ and $B=\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$ which are in Jordan form. However, the matrix $A+B=\begin{pmatrix} 1 & 1\\ 0 & 2\end{pmatrix}$ is not in Jordan form as it's Jordan form is $\begin{pmatrix} 1 & 0 \\ 0 & 2\end{pmatrix}$.

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