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# Jordan canonical form theorem

A Jordan block or Jordan matrix is a matrix of the form

$\begin{pmatrix}\lambda&1&0&\cdots&0\\ 0&\lambda&1&\cdots&0\\ 0&0&\lambda&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&1\\ 0&0&0&\cdots&\lambda\end{pmatrix}$ |

with a constant value $\lambda$ along the diagonal and 1’s on the superdiagonal. Some texts place the 1’s on the subdiagonal instead.

###### Theorem.

Let $V$ be a finite-dimensional vector space over a field $F$ and $t:V\to V$ be a linear transformation. Then, if the characteristic polynomial factors completely over $F$, there will exist a basis of $V$ with respect to which the matrix of $t$ is of the form

$\begin{pmatrix}J_{{1}}&0&\cdots&0\\ 0&J_{{2}}&\cdots&0\\ &&\cdots&\\ 0&0&\cdots&J_{{k}}\end{pmatrix}$ |

where each $J_{{i}}$ is a Jordan block in which $\lambda=\lambda_{{i}}$.

The matrix in Theorem 1 is called a *Jordan canonical form* for the transformation *t*.

Defines:

Jordan block, Jordan matrix

Related:

PartitionedMatrix, SimultaneousUpperTriangularBlockDiagonalizationOfCommutingMatrices, Diagonalizable2

Synonym:

Jordan canonical form

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

15A18*no label found*

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## Corrections

word change and format change for F by Mathprof ✓

reduced Jordan matrix by alozano ✓

typo by IVE ✓

Jordan canonical form by CWoo ✓

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reduced Jordan matrix by alozano ✓

typo by IVE ✓

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