# Jordan canonical form theorem

A 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 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.

 Title Jordan canonical form theorem Canonical name JordanCanonicalFormTheorem Date of creation 2013-03-22 12:59:21 Last modified on 2013-03-22 12:59:21 Owner Mathprof (13753) Last modified by Mathprof (13753) Numerical id 16 Author Mathprof (13753) Entry type Theorem Classification msc 15A18 Synonym Jordan canonical form Related topic PartitionedMatrix Related topic SimultaneousUpperTriangularBlockDiagonalizationOfCommutingMatrices Related topic Diagonalizable2 Defines Jordan block Defines Jordan matrix