Rational Canonical Form

Any square matrix has a canonical form without any need to extend the field of its coefficients. 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)

called the rational canonical form, where 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 product is the characteristic polynomial of .

The rational canonical form is unique, and shows the extent to which the minimal polynomial characterizes a matrix. For example, 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 a -module, that is a module over the ring of polynomials with coefficients in the field . The vector space determines the field , which can be taken to be the maximal field containing 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 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)

To construct a basis for the rational canonical form, it is necessary to write as

(8)

and that is done by finding an appropriate basis for and for . Such a basis is found by determining matrices and that are invertible matrices having entries in (and whose inverses are also in ) such that

(9)

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,

(10)

where is an arbitrary polynomial in . Setting ,

(11)

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

(12)