Editing Frobenius normal form (section)

{{Short description|Canonical form of matrices over a field}}
In [[linear algebra]], the '''Frobenius normal form''' or  '''rational canonical form''' of a [[square matrix|square]] [[matrix (mathematics)|matrix]] ''A'' with entries in a [[field (mathematics)|field]] ''F'' is a [[canonical form]] for matrices obtained by conjugation by [[invertible matrices]] over ''F''. The form reflects a minimal decomposition of the [[vector space]] into [[linear subspace|subspaces]] that are cyclic for ''A'' (i.e., [[linear span|spanned]] by some vector and its repeated [[image (mathematics)|image]]s under ''A''). Since only one normal form can be reached from a given matrix (whence the "canonical"), a matrix ''B'' is [[matrix similarity|similar]] to ''A'' if and only if it has the same rational canonical form as ''A''. Since this form can be found without any operations that might change when [[field extension|extending]] the field ''F'' (whence the "rational"), notably without [[factorization of polynomials|factoring polynomials]], this shows that whether two matrices are similar does not change upon field extensions. The form is named after German mathematician [[Ferdinand Georg Frobenius]].

Some authors use the term rational canonical form for a somewhat different form that is more properly called the '''primary rational canonical form'''. Instead of decomposing into a minimum number of cyclic subspaces, the primary form decomposes into a maximum number of cyclic subspaces. It is also defined over ''F'', but has somewhat different properties: finding the form requires factorization of polynomials, and as a consequence the primary rational canonical form may change when the same matrix is considered over an extension field of ''F''. This article mainly deals with the form that does not require factorization, and explicitly mentions "primary" when the form using factorization is meant.