Jordan Canonical Form

Statement

Let $F$ be an algebraically closed field and $A$ an $n \times n$ matrix over $F$. Then there exists an invertible matrix $P$ and a block-diagonal matrix

such that $A = P J P^{-1}$. Each Jordan block is

The form is unique up to reordering of blocks.

Visualization

Consider $A = \begin{pmatrix} 2 & 1 \\ 0 & 2 \end{pmatrix}$ vs $B = \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}$.

  Matrix A (non-semisimple)        Jordan form of A
  +---------+                      +---------+
  | 2  1    |   ~ similar ~        | 2  1    |  (IS the Jordan block J_2(2))
  | 0  2    |                      | 0  2    |
  +---------+                      +---------+
  eigenvalue λ=2 (double),         one Jordan block of size 2
  minimal poly = (x-2)^2

  Matrix B (semisimple / diagonal) Jordan form of B
  +---------+                      +---------+
  | 2  0    |   ~ similar ~        | 2  0    |  (two blocks J_1(2))
  | 0  2    |                      | 0  2    |
  +---------+                      +---------+
  eigenvalue λ=2 (double),         minimal poly = (x-2), squarefree

The key distinction: $A$ needs a $2 \times 2$ block because $(A - 2I)^2 = 0$ but $(A - 2I) \neq 0$; $B$ is already diagonal so each Jordan block has size $1$.

Matrix	Min poly	Jordan block sizes for $\lambda=2$
$A$	$(x-2)^2$	$\{2\}$
$B$	$x-2$	$\{1, 1\}$

Proof Sketch

1. Generalised eigenspaces. For each eigenvalue $\lambda_i$, the generalised eigenspace $V_i = \ker(A - \lambda_i I)^{n}$ is $A$-invariant. The Cayley–Hamilton theorem guarantees $V = \bigoplus_i V_i$.
2. Jordan basis on each block. On $V_i$, the restriction $N_i = (A - \lambda_i I)|_{V_i}$ is nilpotent. A nilpotent operator on a finite-dimensional space admits a basis of Jordan chains — sequences $v, Nv, N^2 v, \ldots$ that yield one Jordan block per chain.
3. Assemble. Concatenate the Jordan bases across all generalised eigenspaces to get $P$; $J$ is block-diagonal by construction.
4. Uniqueness. The block sizes for $\lambda$ are determined by $\dim \ker (A - \lambda I)^k$ for $k = 1, 2, \ldots$, which are similarity invariants.

Connections

• Cayley–Hamilton TheoremCayley–Hamilton Theorem$$p_A(A) = 0 \;\text{where}\; p_A(\lambda) = \det(\lambda I - A)$$Every square matrix satisfies its own characteristic polynomialRead more → — guarantees $V = \bigoplus_i V_i$ over algebraically closed fields
• Minimal PolynomialMinimal Polynomial$$\mu_A \mid p \iff p(A) = 0$$The minimal polynomial of a matrix is the monic polynomial of least degree that annihilates it.Read more → — block sizes for $\lambda_i$ equal the size of the largest Jordan block, matching the multiplicity of $(\lambda_i)$ in the minimal polynomial
• Spectral TheoremSpectral Theorem$$A = Q \Lambda Q^T \;\text{(real symmetric case)}$$Every real symmetric matrix is orthogonally diagonalizableRead more → — the real/complex spectral theorem gives the semisimple part; Jordan form extends this to the full nilpotent decomposition

import Mathlib.LinearAlgebra.JordanChevalley

/-- The Jordan–Chevalley-Dunford decomposition: every endomorphism of a
    finite-dimensional vector space over a perfect field splits as nilpotent
    plus semisimple.  This is the structural engine behind Jordan canonical form.
    Mathlib's `Module.End.exists_isNilpotent_isSemisimple` is the direct alias. -/
theorem jordan_chevalley_decomp
    {K : Type*} [Field K] [PerfectField K]
    {V : Type*} [AddCommGroup V] [Module K V] [FiniteDimensional K V]
    (f : V →ₗ[K] V) :
    ∃ᵉ (n ∈ Algebra.adjoin K {f}) (s ∈ Algebra.adjoin K {f}),
      IsNilpotent n ∧ Module.End.IsSemisimple s ∧ f = n + s :=
  Module.End.exists_isNilpotent_isSemisimple