Rational Canonical Form

The rational canonical form of a square matrix is a canonical form that does not require extending the field of its coefficients. It is also called Frobenius form or Frobenius normal form (Geck 2020). For instance, if the entries of are rational numbers, then so are the entries of its rational canonical form. The Jordan canonical form may require complex numbers. There exists a nonsingular matrix such that

(1)

where the matrix on the right is the rational canonical form and is the companion matrix for the monic polynomial

(2)

The polynomials are called the "invariant factors" of , and satisfy for , ..., (Hartwig 1996). The polynomial is the matrix minimal polynomial and the characteristic polynomial of is .

The rational canonical form is unique over the specified field. For example, over the rational numbers, there is only one matrix whose matrix minimal polynomial is , which is

(3)

in rational canonical form.

Given a linear transformation , the vector space becomes an -module, that is a module over the ring of polynomials with coefficients in the field . The vector space is finite-dimensional over a specified field , which must contain the entries of a matrix for . The polynomial acts on a vector by . The rational canonical form corresponds to writing as

(4)

where is the ideal generated by the invariant factor in , the canonical form for any finitely generated torsion module over a principal ring such as .

More constructively, given a basis for , there is a module homomorphism

(5)

which is a surjection, given by

(6)

Letting be the module kernel,

(7)

The module kernel is a free module of rank . To construct a vector basis for the rational canonical form, choose bases for and so that the inclusion of has diagonal entries . Such a basis is found by determining matrices and that are invertible matrices having entries in (and whose inverses are also in ) such that

(8)

where is the identity matrix and denotes a diagonal matrix. They can be found by using elementary row and column operations.

The above matrix sends a basis for , written as an -tuple, to an -tuple using a new basis for , and gives the linear transformation from the original basis to the one with the . In particular,

(9)

where is an arbitrary polynomial in . Setting ,

(10)

In particular, is the subspace of which is generated by , where here is the polynomial degree of . Therefore, a vector basis that puts into rational canonical form, with the polynomial degree of , is given by

(11)