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Jordan form of an endomorphism

We call $ \lambda $-Jordan cell a matrix $ J_{\lambda} $ 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 ] \) $
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.

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