If I compute the right Jordan canonical decomposition of a matrix $A$, i.e. I get $$A = VJV^{-1},$$ will I be able to get the left canonical form from it without any further computation?
The right Jordan canonical form is the when where the diagonal right above the diagonal of $J$ is made by zero or one. The left one is the same but the diagonal made by zero or one is the one right below the diagonal of $J$.
$$ \left( \begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array} \right) \left( \begin{array}{cccc} \lambda & 1 & 0 & 0 \\ 0 & \lambda & 1 & 0 \\ 0 & 0 & \lambda & 1 \\ 0 & 0 & 0 & \lambda \\ \end{array} \right) \left( \begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array} \right) = \left( \begin{array}{cccc} \lambda & 0 & 0 & 0 \\ 1 & \lambda & 0 & 0 \\ 0 & 1 & \lambda & 0 \\ 0 & 0 & 1 & \lambda \\ \end{array} \right) $$
