---
tag: 0Z3GDY89T5
kind: statement
role: theorem
name: Decomposition into generalised eigenspaces
msc: [15A18, 15A21]
uses: [07PNDWA7CE, 041JTZCK9Q]
sources:
  - url: https://en.wikipedia.org/wiki/Generalized_eigenvector
    locator: Jordan normal form
---

## 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|0AJ035ZTZE]] 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|0F765KFC2J]].
This [[direct sum|0XYYT5KNTX]] decomposition is the first step towards the [[Jordan normal form|085HQ5G5TE]].

## 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.