Reordering Jordan Matrix Sub-Blocks

Question

Let $J$ and $K$ be $n × n$ Jordan matrices which can be made equal after a reordering of sub-blocks.
Show that $J= S^{-1}KS$ by explaining how to construct $S$.

I understand that these two matrices are similar because changing the order of sub-blocks preserves geometric and algebraic multiplicity of the shared eigenvalues, so we simply have to rearrange those blocks to change between Jordan matrices. However, I don't know how to actually construct the invertible matrix that would accomplish this reordering.

Thanks for any help!

Answer 1

It’s not different from the general case. If you know K and J thus you can calculate S by linear system.

Here is a simple example:

Find the jordan basis for $A$