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 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