Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 3 / 5 / i / Solution

The Jordan–Chevalley decomposition is characterized byOver $\mathbb{C}$, a semisimple linear operator is a diagonalizable one. These conditions determine the two parts uniquely. Concretely, on the generalized eigenspace for an eigenvalue $\lambda$, the semisimple part is $\lambda I$ and the nilpotent part is $X-\lambda I$; both are polynomials in $X$. To see uniqueness, any commuting candidate parts preserve each generalized eigenspace; diagonalize the proposed semisimple part there, and nilpotence then forces its eigenvalue to be the sole eigenvalue $\lambda$ of $X$ on that space.

For the displayed matrix, $\det (tI-X)=(t-1{)}^2$ andSince $I$ is semisimple and commutes with $N$, the required parts are