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The Payoff — Similarity Over Any Field

The complete classification: two matrices are similar exactly when their rational canonical forms agree — and remarkably, this verdict does not depend on which field you compute over.

A complete similarity invariant

Two matrices A and B are similar (B = P A P^-1 for some invertible P) if and only if they have the same rational canonical form — equivalently, the same list of invariant factors. The RCF is a complete invariant: it sees everything that similarity cares about and nothing it doesn't.

This is strictly more powerful than checking the characteristic polynomial and minimal polynomial alone. Those two can coincide for non-similar matrices; the full invariant-factor list never lies.

Similarity is field-independent

Here is the deep surprise. Suppose A and B have rational entries. They might become similar only after you enlarge Q to the complex numbers — or so you'd fear. In fact similarity is field-independent: if A and B are similar over a big field, they are already similar over the small field they live in.

The reason is mechanical and beautiful: the Smith normal form of xI - A is computed by polynomial row/column operations that never leave the base field. So the invariant factors of a rational matrix are themselves rational, and enlarging the field cannot change them.

How rational and Jordan forms meet

When the field is big enough to contain every eigenvalue (e.g. over C), both forms exist and they are two views of the same elementary divisors — see rational form vs Jordan form. A prime-power elementary divisor (x - lambda)^m becomes a Jordan block of size m with eigenvalue lambda, or the companion matrix of (x - lambda)^m in the primary rational form.

Same operator, two canonical forms when lambda lives in the field:

  elementary divisor   (x - lambda)^3

  Jordan block:          companion (primary rational):
  [ lambda  1     0    ]   [ 0  0   lambda^3            ]
  [ 0     lambda  1    ]   [ 1  0  -3 lambda^2          ]
  [ 0       0   lambda ]   [ 0  1   3 lambda            ]

Same invariant data; different bases. Over a field missing lambda,
only the companion (rational) form survives.
One elementary divisor, rendered as a Jordan block or as a companion block.

That is the arc's payoff. Starting from a first-course question — when are two matrices the same operator in disguise? — we passed through modules, the Smith normal form, and invariant factors to reach a single, computable, field-independent answer that needs no eigenvalues at all. The rational canonical form is the final word on similarity.