---
tag: 085HQ5G5TE
kind: statement
role: theorem
name: Jordan normal form
aliases: [jordan-canonical-form]
msc: [15A21, 15A20]
uses: [0PBFS92AJP, 027KDW2K72, 0M0SNKZ482]
sources:
  - url: https://en.wikipedia.org/wiki/Jordan_normal_form
    locator: Overview
same_as:
  wikidata: Q838495
  wikipedia: Jordan normal form
---

## Statement

Let $F$ be an algebraically closed field, for example $\C$, and let $A$ be an $n \times n$ matrix over $F$.
Then $A$ is [[similar|0E89HQATM8]] to a [[block diagonal matrix|001T5FJDC1]]
$$J = \begin{pmatrix} J_{k_1}(\lambda_1) & & \\ & \ddots & \\ & & J_{k_r}(\lambda_r) \end{pmatrix},$$
whose diagonal blocks are [[Jordan blocks|07PH4RX0X5]].
The matrix $J$ is unique up to the order of the blocks.
It is the *Jordan normal form* of $A$.

## Remarks

The $\lambda_i$ are the eigenvalues of $A$; they need not be distinct.
The number of blocks for $\lambda$ is its geometric multiplicity, and the sum of their sizes is its algebraic multiplicity.
The largest block for $\lambda$ has size equal to the multiplicity of $\lambda$ in the minimal polynomial.
The theorem holds over any field $F$ for the matrices whose characteristic polynomial splits over $F$.

## Explanations

### L1 human

Every complex square matrix becomes almost diagonal in a good basis.
The only entries off the diagonal are some $1$s just above it, inside the Jordan blocks.
Two complex matrices are similar exactly when they have the same Jordan form, up to the order of the blocks.