Endomorphism/Trigonalizable/Direct sum/Fact/Proof

Proof

Let

${\chi }_{\phi }=(X-{\lambda }_1{)}^{k_1}\cdots (X-{\lambda }_m{)}^{k_m}$

be the characteristic polynomial, which splits into linear factors according to fact, where the ${\lambda }_i$ are different. We do induction over $m$. For $m=1$, there is only one eigenvalue $\lambda$ and only one generalized eigenspace. Due to fact, the minimal polynomial is of the form $(X-\lambda {)}^s$ and thus $V={\mathrm{GeEig}}_{\lambda }(\phi )$. Suppose that the statement is already proven for smaller $m$. We set $P=(X-{\lambda }_1{)}^{k_1}$ and $Q=(X-{\lambda }_2{)}^{k_2}\cdots (X-{\lambda }_m{)}^{k_m}$. We are then in the situation of fact and fact. Therefore, we have a direct sum decomposition into $\phi$-invariant linear subspaces

$V={\mathrm{GeEig}}_{{\lambda }_1}(\phi )\oplus \mathrm{kern}Q(\phi ).$

The characteristic polynomial is, according to fact, the product of the characteristic polynomials of the restrictions to the spaces. Because of fact, the polynomial $(X-{\lambda }_1{)}^{k_1}$ is the characteristic polynomial of the restriction to the first generalized eigenspace, hence, $Q$ is the characteristic polynomial of the restriction to $\mathrm{kern}Q(P)$. In particular, this restriction is also trigonalizable. By the induction hypothesis, $\mathrm{kern}Q(P)$ is the direct sum of the generalized eigenspaces for ${\lambda }_2,\dots ,{\lambda }_m$. Altogether, this implies the direct sum decomposition of $V$ and of $\phi$.

To fact