I need some help with exercise 4 in section 57 of Paul Halmos' Finite-Dimensional Vector Spaces:
If $A$ is a linear transformation (on a finite-dimensional vector space over an algebraically closed field), then there exist linear transformation $B$ and $C$ such that $A = B + C$, $B$ is diagonable, $C$ is nilpotent, and $BC = CB$; the transformations $B$ and $C$ are uniquely determined by these conditions.
My thoughts:
Let $T = X^{-1}AX$ such that $T$ is "triangular". Then $T = D + N$, where $D$ is a diagonable and $N$ is nilpotent and $A = XTX^{-1} = XDX^{-1} + XNX^{-1}$.