Jordan form of an endomorphism
Posted December 15th, 2007 by Structure
We call 
-Jordan cell a matrix 
which has a particular form.
In 4-dimension we have
![$ J_{\lambda}=\left [\begin{array}{cccc}<br />
\lambda&0&0&0\\<br />
1&\lambda&0&0\\<br />
0&1&\lambda&0\\<br />
0&0&1&\lambda<br />
\end {array}\right ] \) $](/files/tex/98a736089ebe3659f36ae3f212c45e0a28ae6df2.png "$ J_{\lambda}=\left [\begin{array}{cccc}<br />
\lambda&0&0&0\\<br />
1&\lambda&0&0\\<br />
0&1&\lambda&0\\<br />
0&0&1&\lambda<br />
\end {array}\right ] \) $")
We say that an endomorphism is in Jordan form if there is a basis of the vector space in which the matrix associated to endomorphism is a "diagonal "matrix of Jordan cells.
Necessary and sufficient condition for an endomorphism to admit a Jordan form is that the characteristic polynomial has n solution (eigenvalues for endomorphism) in field k.