Where Vol I left off
In a first course you learned diagonalization: if a matrix A has a full set of independent eigenvectors, you can write A = P D P^-1 with D diagonal. Not every matrix is diagonalizable, so Jordan form patched the gap by allowing 1's just above the diagonal. Both tools answer the same question — when are two matrices similar? — but both have a hidden cost.
The hidden cost is eigenvalues. Jordan blocks are built from the roots lambda of the characteristic polynomial. If those roots do not live in your field, there is no Jordan form over that field at all.
A matrix with no eigenvalues
Work over the rationals Q. Take the rotation-by-90-degrees matrix. Its characteristic polynomial is x^2 + 1, which has no root in Q (or even in R). So over Q this matrix is not diagonalizable and has no Jordan form — yet it is a perfectly ordinary linear operator that we ought to be able to classify.
A = [ 0 -1 ] over the field Q
[ 1 0 ]
char poly det(xI - A) = x^2 + 1
roots in Q? none (x^2 + 1 is irreducible over Q)
=> A is NOT diagonalizable over Q
=> A has NO Jordan form over Q
but A is a genuine operator -- we still want a canonical form for it.The form that needs no roots
The rational canonical form (RCF) solves exactly this. It is built not from eigenvalues but from polynomials with coefficients in your field — so it exists for every operator over every field, no algebraic closure required. The price is that its blocks look different: instead of near-diagonal Jordan blocks, you get companion matrices of polynomials.