Prove two Jordan matrices with Jordan blocks in different order are similar

Question

Suppose we have two $n \times n$ Jordan matrices which only differ in the order of Jordan blocks. I want to show that these Jordan matrices are similar. In general, I understand how to get to this conclusion. Let $A, B$ be the mentioned Jordan matrices. To transform $B$ to $A$ we have to move the Jordan blocks around. We first multiply $P_1B$ where $P_1$ is the permutation matrix moving the rows of $B$ into their respective positions in $A$. Then we multiply $(P_1B)P_2$ where $P_2$ rearranges the columns to fit those in $A$. We notice $P_1P_2=I$ and hence $A=C^{-1}BC$ and the matrices are similar. My problem here is that my attempts to formally write down the fact that $P_1P_2=I$ are extremely clunky, with lots of indices. Is there an elegant way to formulate this conclusion?

Answer 1

You could view this as computing the decomposition of the linear operator represented by $A$ and $B$. We can decompose these linear operators into a direct sum of quotient modules, each one representing a block in the JCF. The ordering is unimportant so they will have the same decomposition, and thus be similar. This can elaborate a bit more.