Editing Algebraically closed field (section)

===Every endomorphism of ''F<sup>n</sup>'' has some eigenvector===
The field ''F'' is algebraically closed if and only if, for each natural number ''n'', every [[linear map]] from ''F<sup>n</sup>'' into itself has some [[eigenvector]].

An [[endomorphism]] of ''F<sup>n</sup>'' has an eigenvector if and only if its [[characteristic polynomial]] has some root. Therefore, when ''F'' is algebraically closed, every endomorphism of ''F<sup>n</sup>'' has some eigenvector. On the other hand, if every endomorphism of ''F<sup>n</sup>'' has an eigenvector, let ''p''(''x'') be an element of ''F''[''x'']. Dividing by its leading coefficient, we get another polynomial ''q''(''x'') which has roots if and only if ''p''(''x'') has roots. But if {{math|1=''q''(''x'') = ''x<sup>n</sup>'' + ''a''<sub>''n''&thinsp;&minus;&thinsp;1</sub>&thinsp;''x''<sup>''n''&thinsp;&minus;&thinsp;1</sup>&thinsp;+ ⋯ + ''a''<sub>0</sub>}}, then ''q''(''x'') is the characteristic polynomial of the ''n×n'' [[companion matrix]]
:<math>\begin{pmatrix}
  0 & 0 & \cdots & 0 & -a_0\\
  1 & 0 & \cdots & 0 & -a_1\\
  0 & 1 & \cdots & 0 & -a_2\\
  \vdots & \vdots & \ddots & \vdots & \vdots\\
  0 & 0 & \cdots & 1 & -a_{n-1}
\end{pmatrix}.</math>