conjugation of operators

Conjugation is the operation A -> P^-1 A P viewed not as a one-off formula but as something a group does. Fix an invertible operator P; conjugating by P is the map that sends every operator A to P^-1 A P. As P ranges over the whole general linear group GL(V), you get an action of GL(V) on the set of operators, and the orbits of that action are exactly the similarity classes.

This group-action viewpoint reorganizes everything. Saying A and B are similar is saying they lie in the same orbit. Asking for a canonical form is asking for one chosen representative of each orbit. Asking which quantities are similarity invariants is asking which functions on operators are constant along orbits — the trace, determinant, characteristic and minimal polynomials, and so on are precisely these invariant functions.

Conjugation respects the algebra structure: it is an automorphism of End(V). It preserves sums, scalar multiples, and crucially products, since P^-1(AB)P = (P^-1 A P)(P^-1 B P). So conjugating is relabeling the space by P and then watching the same operators act — composition, powers, and polynomial identities all carry over. This is why an invariant like the minimal polynomial cannot change: it is an algebraic relation, and algebra is preserved.

The same idea recurs throughout mathematics: conjugation in a group, change of variables in a differential operator, similarity transformations in physics where it implements a change of reference frame. In every case the moral is identical — conjugation is a change of viewpoint, and the things worth naming are the ones it leaves alone.