Explain how the structure theorem for finitely-generated modules over a PID applies to a linear operator on a finite dimensional vector space.
Solution
Let \(V\) be a finite-dimensional vector space over a field \(F\) and let \(T\in\operatorname{End}_F(V)\).
1. Regard \(V\) as an \(F[x]\)-module by\[ f(x)\cdot v=f(T)v. \]
Polynomial evaluation respects addition and multiplication: \[ (f+g)(T)=f(T)+g(T), \qquad (fg)(T)=f(T)g(T), \qquad 1(T)=I. \] Thus the displayed rule satisfies the module axioms.
2. This \(F[x]\)-module is finitely generated and torsion.
Any \(F\)-basis of \(V\) generates \(V\) as an \(F[x]\)-module, so it is finitely generated. By Cayley–Hamilton, the characteristic polynomial satisfies \[ \chi_T(T)=0, \] so every vector is annihilated by the nonzero polynomial \(\chi_T(x)\). Hence the module is torsion.
3. Since \(F[x]\) is a PID, the structure theorem gives monic polynomials\[ a_1(x)\mid a_2(x)\mid\cdots\mid a_r(x) \]such that\[ V\cong\bigoplus_{i=1}^r F[x]/(a_i(x)) \]as \(F[x]\)-modules.
2. The \(a_i\) are the invariant factors of the module, hence of the operator \(T\).
4. On the summand \(F[x]/(a_i)\), multiplication by \(x\) is represented by the companion matrix of \(a_i\).
If \(d=\deg a_i\), then the residue classes \[ 1,x,\ldots,x^{d-1} \] form an \(F\)-basis of \(F[x]/(a_i)\). Multiplication by \(x\) shifts this basis, and the relation \(a_i(x)=0\) expresses \(x^d\) as the corresponding linear combination of lower powers. This is exactly the companion matrix \(C(a_i)\).
5. Therefore \(T\) is similar to the rational canonical form\[ C(a_1)\oplus\cdots\oplus C(a_r). \]
3. Since multiplication by \(x\) is exactly the action of \(T\), its matrix is the displayed block diagonal matrix.
6. The invariant factors recover the characteristic and minimal polynomials:\[ \chi_T(x)=\prod_{i=1}^r a_i(x), \qquad \mu_T(x)=a_r(x). \]
The characteristic polynomial of a companion matrix \(C(a_i)\) is \(a_i\), so block diagonality gives the product formula. A polynomial annihilates the direct sum exactly when it is divisible by every \(a_i\); because \(a_1\mid\cdots\mid a_r\), their least common multiple is \(a_r\). Hence the minimal polynomial is \(a_r\).
7. Factoring the invariant factors into powers of irreducibles gives the elementary-divisor form; after passing to a splitting field, this refines to Jordan blocks.
The primary decomposition part of the PID-module structure theorem decomposes each cyclic module according to the prime-power factors of the \(a_i\). Over a splitting field those irreducibles are linear, so the cyclic modules \(F[x]/((x-\lambda)^e)\) correspond to Jordan blocks of size \(e\) with eigenvalue \(\lambda\).