Frobenius Normal Form - General Case and Theory

General Case and Theory

Fix a base field F and a finite-dimensional vector space V over F. Given a polynomial p(x) ∈ F, there is associated to it a companion matrix C whose characteristic polynomial is p(x).

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-module with the action of x given by A and extending by linearity) satisfies the F-module isomorphism

V ≅ F/(a₁(x)) ⊕ … ⊕ F/(a_n(x))

where the a_i(x) ∈ F may be taken to be non-units, unique as monic polynomials, and can be arranged to satisfy the relation

a₁(x) | … | a_n(x)

where "a | b" is notation for "a divides b".

Sketch of Proof: Apply the structure theorem for finitely generated modules over a principal ideal domain to V, viewing it as an F-module. Note that any free F-module is infinite-dimensional over F, so that the resulting direct sum decomposition has no free part since V is finite-dimensional. The uniqueness of the invariant factors requires a separate proof that they are determined up to units; then the monic condition ensures that they are uniquely determined. The proof of this latter part is omitted. See for details.

Given an arbitrary square matrix, the elementary divisors used in the construction of the Jordan normal form do not exist over F, so the invariant factors a_i(x) as given above must be used instead. These correspond to factors of the minimal polynomial m(x) = a_n(x), which (by the Cayley–Hamilton theorem) itself divides the characteristic polynomial p(x) and in fact has the same roots as p(x), not counting multiplicities. Note in particular that the Theorem asserts that the invariant factors have coefficients in F.

As each invariant factor a_i(x) is a polynomial in F, we may associate a corresponding block matrix C_i which is the companion matrix to a_i(x). In particular, each such C_i has its entries in the field F.

Taking the matrix direct sum of these blocks over all the invariant factors yields the rational canonical form of A. Where the minimal polynomial is identical to the characteristic polynomial, 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.