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Here are two problems about additive Jordan Decomposition, that is, $V$ is a finite dimension complex vector space, for a linear map $T\in End (V)$, we have $T=D+N$, and $DN=ND$, where $D$ is diagonalizable and $N$ is nilpotent. The decomposition is unique.

Suppose $V$ is a finite dimensional complex vector space and $u,v\in End(V)$ such that $u\circ v=v\circ u$. How can I express the Jordan decomposition of $u\circ v$ in terms of the Jordan decomposition of $u$ and $v$?

Zijie Tang
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  • Please only one question per post. Better remove your Problem 1 and ask for clarification in comment below the anwer to the linked post ("but I'm still confused": explain there what confuses you), and focus here on your Problem 2, and avoid "no clue" questions. – Anne Bauval Jan 05 '25 at 16:49
  • over an algebraically closed field like $\mathbb C$, $ad(u)$ is diagonalizable iff $u$ is. For the direction you're seeking in Problem 1: https://math.stackexchange.com/questions/1345661/eigenvalues-of-linear-operator-fa-ab-ba ; the accepted answers shows $n^2$ linearly independent eigenvectors [albeit with eigenvalues $\lambda_i -\lambda_j$ in your case]; the other direction of the iff is https://math.stackexchange.com/questions/4887248/if-fx-ax-xa-is-diagonalizable-show-that-a-is-diagonalizable/ – user8675309 Jan 05 '25 at 18:40

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Here is an answer to your Problem 2 (better ask only 1 question per post, so remove your Problem 1 from here and post it separately, or ask for clarifications by comments in Jordan Decomposition, explaining what confuses you, but first think about @user1551 and @user8675309's hints above).

Let $u=D_u+N_u$ and $v=D_v+N_v$ be the additive Jordan decompositions of $u$ and $v$. Since $u$ and $v$ commute, the four linear maps $$D_u,\quad N_u,\quad D_v,\quad N_v$$ are pairwise commuting, because they can be expressed as polynomials in $u$ and $v$ respectively. Therefore:

  • $N:=N_u\circ D_v+D_u\circ N_v+N_u\circ N_v$ is nilpotent,
  • $D:=D_u\circ D_v$ is diagonalizable (because $D_u,D_v$ are simultaneaously diagonalizable), and
  • $N,D$ commute.
Anne Bauval
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