Suppose $A$ is a linear transformation of a 3-dim vector space $V$, defined as $A(\epsilon_1,\epsilon_2,\epsilon_3)=(\epsilon_1,\epsilon_2,\epsilon_3) \begin{pmatrix} -10 & 12 & 7\\ -3 & 4 & 2\\ -13 & 15 & 9 \end{pmatrix}, $here $\{\epsilon_i\}$ is a basis of $V$.
Is there a concise way to find the transition matrix to a new basis under which the linear operator $A$ has a matrix of Jordan form? And what's behind the solution?