In the Jordan-Chevalley decomposition $M=D+N$, how obtaining $D$ and $N$ as polynomials in $M$

Question

The Jordan-Chevalley expresses a linear operator $M$ as
$$
M = D + N,
$$
where $D$ is semisimple (diagonalizable), $N$ is nilpotent and $DN=ND$. Although it is stated in many sources that $D$ and $N$ can be written as polynomials in $M$, I never see any method for obtaining such polynomials. I want to emphasize that I am not interested in a proof about the existence of these polynomials, but in a procedure for explicitly obtaining them given the Jordan-Chevalley decomposition.

Answer 1

Write $\chi_M(X)=\prod_{k=1}^s{(X-\lambda_k)^{\alpha_k}}$ (characteristic polynomial).

From among others Cayley-Hamilton, we know that $\bigoplus_{k=1}^s{\ker((M-\lambda_kI)^{\alpha_k})}=\mathbb{C}^n$ (for instance).

Let, for each $k$, $A_k$, $B_k$ be polynomials such that $A_k(X)(X-\lambda_k)^{\alpha_k}+B_k(X)\prod_{l \neq k}{(X-\lambda_l)^{\alpha_l}}=1$.

Let $P_k(X)=B_k(X)\prod_{l \neq k}{(X-\lambda_l)^{\alpha_l}}$.

Then, if $(M-\lambda_kI)^{\alpha_k}v=0$, then $P_k(M)v=v$. On the other hand, if $l \neq k$ and $(M-\lambda_lI)^{\alpha_l}v=0$, then $P_k(M)v=0$.

Then $D=\sum_k{\lambda_kP_k(M)}$.