Generalized eigenspaces for distinct eigenvalues form a direct sum
= Generalized eigenspaces for distinct eigenvalues form a direct sum
For a linear operator $f$, the generalized eigenspaces
$$
\ker(f-\alpha I)^n
$$
belonging to distinct eigenvalues are linearly independent. On the generalized $\alpha$-eigenspace, $f-\beta I=(\alpha-\beta)I+N$ is invertible for $\beta\ne\alpha$ because the nilpotent part $N$ has a finite geometric-series inverse.