I have been reading Denis Serre's book Matrices Theory and Applications(GTM216). In the proof of Dunford-Taylor formula in page 96, the author said that it is enough to assume $A=\lambda I+N,$ where $N$ is a nilpotent. I am confused since for any matrix $M$ defined on $\mathbb{C},$ it can decompose as $M=D+N,$while $D$ is diagonalizable and $N$ is nilpotent, but the eigenvalues of $D$ are not necessarily all the same. So how can I assume $A=\lambda I+N?$
The following is proof of Dunford-Talyor formula, and the conjugation property is "$f(M^{-1}AM)=M^{-1}f(A)M$", the proposition 3.20 is "for matrix $M$ defined on $\mathbb{C}$,then $M$ decomposes as a sum $M=D+N,$ where $D$ is diagonalizable, $N$ is nilpotent, $DN=ND$ and spectrum of $D$ equals spectrum of $M$."
