Decomposition into generalised eigenspaces

Statement

Let $F$ be a field, let $V$ be a finite-dimensional vector space over $F$, and let $T \colon V \to V$ be a linear map. Assume that the characteristic polynomial of $T$ splits over $F$ as $p_T(t) = (t - \lambda_1)^{m_1} \cdots (t - \lambda_k)^{m_k}$ with pairwise distinct $\lambda_i$. Then $$V = V_{\lambda_1} \oplus \cdots \oplus V_{\lambda_k},$$ where $V_{\lambda_i}$ is the generalised eigenspace of $\lambda_i$, and $\dim V_{\lambda_i} = m_i$.

Remarks

The hypothesis holds for every $T$ when $F$ is algebraically closed, for example $F = \C$. Each $V_{\lambda_i}$ is $T$-invariant, and $T - \lambda_i \id$ is nilpotent on it. A proof uses the coprime polynomials $(t - \lambda_i)^{m_i}$ and the Cayley–Hamilton theorem. This direct sum decomposition is the first step towards the Jordan normal form.

Explanations

L1 human

Over $\C$, the space splits into pieces, one for each eigenvalue. On each piece, the map is $\lambda$ times the identity plus a nilpotent part. So a hard problem breaks into simpler problems, one for each eigenvalue.