Let $\{J_{n_1}(\lambda_1),...,J_{n_m}(\lambda_m)\}$ and $\{J_{l_1}(\mu_1),...,J_{l_k}(\mu_k)\}$ be finite sequences of Jordan blocks with entries in a field $F$.
Let $A\triangleq J_{n_1}(\lambda_1)\oplus \ldots \oplus J_{n_m}(\lambda_m)$ and $B\triangleq J_{l_1}(\mu_1)\oplus \ldots \oplus J_{l_k}(\mu_k)$.
Assuming $A$ and $B$ are similar, how do I prove that there exists a bijection $f:\{1,\dots,m\}\rightarrow \{1,\dots,k\}$ such that $J_{n_i}(\lambda_i) = J_{l_{f(i)}}(\mu_{f(i)})$?
One should construct a permutation to prove this sentence.
This is obviously true, but I find it extremely tedious and bit hard to write down an arbitrary permutation in first-order logic. (this is why I don't really like matrix-theory..even though it is so useful).
How do i prove this rigorously?