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. The polynomials ψ_i are called the "invariant factors" of T, and satisfy ψ_i |ψ_(i + 1) for i = 1, ..., s - 1. The polynomial ψ_s is the matrix minimal polynomial and the product product ψ_i is the characteristic polynomial of T.
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