Imagine I have the following matrix in Jordan form \begin{align} \begin{pmatrix} 3&1&0\\ 0&3&0\\ 0&0&2 \end{pmatrix} \end{align} By defining this matrix to be a representation of a linear map in a basis $B$, we can imagine infinite number of linear maps being generated from this matrix, by choosing different $B$. Let us call two of them $T_1$ and $T_2$.
Now $T_1$ and $T_2$ will have almost identical properties (eigenvalues with the same multiplicities, characteristic equation, minimal polynomial), because they have the same canonical form. The only difference, I think, would be eigenvectors since this representation is in different bases.
Is that the only different? and how are all these linear maps related to each other? can we say that $T_1 = S^{-1}T_2S$ for some invertible linear map $S$?