# A function that maps matrix to its Jordan form is linear?

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 $$B+D$$?

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 $$\begin{pmatrix} 1 & 0 \\ 0 & 2\end{pmatrix}$$.