Any square matrix T has a canonical form without any need to extend the field of its coefficients. For instance, if the entries of T 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 Q such that Q^(-1) T Q = diag[L(ψ_1), L(ψ_2), ..., L(ψ_s)], called the rational canonical form, where L(f) is the companion matrix for the monic polynomial f(λ) = f_0 + f_1 λ + ... + f_(n - 1) λ^(n - 1) + λ^n.

block diagonal matrix | characteristic polynomial | companion matrix | field | invariant factor | Jordan canonical form | matrix | matrix minimal polynomial | principal ring | similar matrices | Smith normal form