Endomorphism/Trigonalizable/Direct sum/Fact/Proof

Proof

Let

${}\chi _{\varphi }=(X-\lambda _{1})^{k_{1}}\cdots (X-\lambda _{m})^{k_{m}}\,$

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 ${}V=\operatorname {GeEig} _{\lambda }(\varphi )$. 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 ${}\varphi$-invariant linear subspaces

${}V=\operatorname {GeEig} _{\lambda _{1}}(\varphi )\oplus \operatorname {kern} Q(\varphi )\,.$

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 ${}\operatorname {kern} Q(P)$. In particular, this restriction is also trigonalizable. By the induction hypothesis, ${}\operatorname {kern} Q(P)$ is the direct sum of the generalized eigenspaces for ${}\lambda _{2},\ldots ,\lambda _{m}$. Altogether, this implies the direct sum decomposition of ${}V$ and of ${}\varphi$.

To fact