Editing Frobenius normal form (section)

==General case and theory==
Fix a base field ''F'' and a finite-dimensional [[vector space]] ''V'' over ''F''. Given a polynomial ''P'' ∈ ''F''[''X''], there is associated to it a [[companion matrix]] ''C''<sub>''P''</sub> whose characteristic polynomial and minimal polynomial are both equal to ''P''.

'''Theorem''': Let ''V'' be a finite-dimensional vector space over a field ''F'', and ''A'' a square matrix over ''F''. Then ''V'' (viewed as an ''F''[''X'']-[[module (mathematics)|module]] with the action of ''X'' given by ''A'') admits a ''F''[''X'']-module isomorphism

:''V'' ≅ ''F''[''X'']/''f''<sub>1</sub> ⊕ … ⊕ ''F''[''X'']/''f<sub>k</sub>''

where the ''f<sub>i</sub>'' ∈ ''F''[''X''] may be taken to be monic polynomials of positive [[degree of a polynomial|degree]] (so they are non-[[unit (ring theory)|unit]]s in ''F''[''X'']) that satisfy the relations

:''f''<sub>1</sub> | ''f''<sub>2</sub> | … | ''f<sub>k</sub>''

(where "a | b" is notation for "''a'' divides ''b''"); with these conditions the list of polynomials ''f<sub>i</sub>'' is unique.

''Sketch of Proof'': Apply the [[structure theorem for finitely generated modules over a principal ideal domain]] to ''V'', viewing it as an ''F''[''X'']-module. The structure theorem provides a decomposition into cyclic factors, each of which is a [[quotient module|quotient]] of ''F''[''X''] by a proper [[ideal (ring theory)|ideal]]; the zero ideal cannot be present since the resulting [[free module]] would be infinite-dimensional as ''F'' vector space, while ''V'' is finite-dimensional. For the polynomials ''f''<sub>''i''</sub> one then takes the unique monic generators of the respective ideals, and since the structure theorem ensures containment of every ideal in the preceding ideal, one obtains the divisibility conditions for the ''f''<sub>''i''</sub>. See [DF] for details.

Given an arbitrary square matrix, the [[elementary divisors]] used in the construction of the [[Jordan normal form]] do not exist over ''F''[''X''], so the [[invariant factors]] ''f''<sub>''i''</sub> as given above must be used instead. The last of these factors ''f<sub>k</sub>'' is then the minimal polynomial, which all the 
invariant factors therefore divide, and the product of the invariant factors gives the characteristic polynomial. Note that this implies that the minimal polynomial divides the characteristic polynomial (which is essentially the [[Cayley-Hamilton theorem]]), and that every [[irreducible polynomial|irreducible]] factor of the characteristic polynomial also divides the minimal polynomial (possibly with lower multiplicity).

For each invariant factor ''f<sub>i</sub>'' one takes its [[companion matrix]] ''C''<sub>''f<sub>i</sub>''</sub>, and the block diagonal matrix formed from these blocks yields the '''rational canonical form''' of ''A''. When the minimal polynomial is identical to the characteristic polynomial (the case ''k''&nbsp;=&nbsp;1), the Frobenius normal form is the companion matrix of the characteristic polynomial. As the rational canonical form is uniquely determined by the unique invariant factors associated to ''A'', and these invariant factors are independent of basis, it follows that two square matrices ''A'' and ''B'' are similar if and only if they have the same rational canonical form.