What is the connection between the structure theorem for modules over a PID and conjugacy classes in the general linear group over a field?
Solution
Let \(V\) be a finite-dimensional vector space over a field \(F\), and let \(T\in\operatorname{End}_F(V)\).
1. The operator \(T\) makes \(V\) into an \(F[t]\)-module by\[ f(t)\cdot v=f(T)v. \]
Polynomial evaluation at \(T\) is a ring homomorphism \(F[t]\to\operatorname{End}_F(V)\), so this defines an \(F[t]\)-module structure. Conversely, an \(F[t]\)-module structure extending the given \(F\)-vector-space structure is determined by the action of \(t\), hence by one linear operator.
2. Two operators \(S,T\in\operatorname{End}_F(V)\) are conjugate in \(\operatorname{GL}(V)\) iff the corresponding \(F[t]\)-modules are isomorphic.
An \(F\)-linear isomorphism \(P:V\to V\) is \(F[t]\)-linear exactly when \[ P(Tv)=S(Pv) \] for every \(v\), i.e. \[ PT=SP. \] Since \(P\) is invertible, this is equivalent to \(S=PTP^{-1}\).
3. Since \(F[t]\) is a PID and \(V\) is finitely generated and torsion as an \(F[t]\)-module, the structure theorem classifies \(T\) up to conjugacy.
Finite-dimensionality over \(F\) makes \(V\) finitely generated over \(F[t]\). Cayley-Hamilton gives a nonzero polynomial annihilating all of \(V\), so the module is torsion. Hence \[ V\cong\bigoplus_i F[t]/(f_i(t)) \] with invariant factors \[ f_1\mid f_2\mid\cdots\mid f_r. \] These invariant factors determine and are determined by the conjugacy class of \(T\).
4. The companion matrices of the invariant factors give the rational canonical form of \(T\).
On the cyclic module \(F[t]/(f)\), multiplication by \(t\) is represented in the standard cyclic basis by the companion matrix of \(f\). Taking the direct sum over the invariant factors produces a canonical matrix representative of the conjugacy class.
Thus conjugacy classes of linear operators are precisely isomorphism classes of the associated finite torsion \(F[t]\)-modules, and the PID structure theorem is the module-theoretic source of rational canonical form.