Editing Companion matrix (section)

==Similarity to companion matrix==
Any matrix {{mvar|A}} with entries in a [[field (mathematics)|field]] {{mvar|F}} has characteristic polynomial <math> p(x) = \det(xI - A) </math>, which in turn has companion matrix <math> C(p) </math>. These matrices are related as follows.

The following statements are equivalent:
* ''A''  is [[similar (linear algebra)|similar]] over ''F'' to <math> C(p) </math>, i.e. ''A'' can be conjugated to its companion matrix by matrices in GL''<sub>n</sub>''(''F'');
* the characteristic polynomial <math> p(x) </math> coincides with the minimal polynomial of ''A'' , i.e. the minimal polynomial has degree ''n'';
* the linear mapping <math>A:F^n\to F^n</math> makes <math>F^n</math> a [[Cyclic module|cyclic]] <math>F[A]</math>-module, having a basis of the form <math>\{v,Av,\ldots,A^{n-1}v\} </math>; or equivalently <math>F^n \cong F[X]/(p(x))</math> as <math>F[A]</math>-modules.

If the above hold, one says that ''A'' is ''non-derogatory''.

Not every square matrix is similar to a companion matrix, but every square matrix is similar to a [[block diagonal]] matrix made of companion matrices. If we also demand that the polynomial of each diagonal block divides the next one, they are uniquely determined by ''A'', and this gives the [[Frobenius normal form|rational canonical form]] of ''A''.