Engineering / Mathematics — Polynomial Algorithms
The Algebra of Linear Transformations
The endomorphism algebra of a vector space, minimal and characteristic polynomials, and the module view of a linear operator.
Executive summary
Linear transformations of a vector space form an algebra under composition. Viewing the space as a module over the polynomial ring, with X acting as the operator, converts operator questions into module questions.
The minimal polynomial generates the annihilator ideal, and the structure theorem for modules over a principal ideal domain gives the canonical forms.
Learning objectives
- Describe the endomorphism algebra.
- Set up the F[X]-module structure of an operator.
- Relate minimal and characteristic polynomials.
01The endomorphism algebra
Endomorphism algebra
End(V) = Hom(V, V) with composition as multiplication is an F-algebra, isomorphic to the algebra of n × n matrices once a basis is chosen.
It has dimension n² over F and is non-commutative for n ≥ 2. Its units are the invertible transformations, forming the general linear group.
02The module view
Given a transformation T of V, define an F[X]-module structure on V by letting X act as T.
f(X) · v := f(T)(v) for f ∈ F[X], v ∈ VAnnihilator and minimal polynomial
The annihilator of V as an F[X]-module is a non-zero ideal, and its monic generator is the minimal polynomial of T.
Non-zero because End(V) is finite dimensional, so the powers of T are eventually dependent. This is the same argument as for minimal polynomials of algebra elements, and it is the same theorem.
| Operator concept | Module concept |
|---|---|
| Invariant subspace | Submodule |
| Minimal polynomial | Generator of the annihilator |
| Eigenvector | Element annihilated by X − λ |
| Cyclic vector | Generator of the module |
| Canonical form | Decomposition into cyclic modules |
03Minimal and characteristic polynomials
Cayley-Hamilton and divisibility
The characteristic polynomial annihilates T, so the minimal polynomial divides it. Both have the same irreducible factors, differing only in multiplicities.
- Minimal polynomial
degree ≤ nGenerator of the annihilator; smallest annihilating polynomial - Characteristic polynomial
degree exactly ndet(XI − T); always annihilates by Cayley-Hamilton - Equal
when V is cyclicThere is a vector whose T-orbit spans V
The structure theorem for finitely generated modules over a principal ideal domain, applied to F[X], decomposes V into a direct sum of cyclic modules. The invariant factors of that decomposition give the rational canonical form, which requires no field extension.
Over an algebraically closed field the same theorem gives the Jordan form. Over a finite field the rational canonical form is what is available and what computations produce, since the eigenvalues may lie in an extension.
04Frequently asked questions
Why is the module view worth the abstraction?
Because it imports the structure theorem for modules over a principal ideal domain wholesale. The canonical forms of linear algebra are corollaries of a single module theorem rather than separate results requiring separate proofs.
When do the minimal and characteristic polynomials coincide?
Exactly when the module is cyclic — when some vector's orbit under T spans the whole space. This is the generic case, and it is what makes the random projection method for computing minimal polynomials usually succeed.
Does Cayley-Hamilton have a one-line proof?
Not a correct one. The tempting substitution of the matrix into its own characteristic polynomial confuses two different rings. Legitimate proofs use the adjugate identity or a density argument over an algebraically closed field.
Sources and method
Structural reference: Victor Shoup, A Computational Introduction to Number Theory and Algebra, Version 1, Cambridge University Press, 2005 — book pages 440-447.
This page carries the durable method layer only: definitions, constructions, algorithms, complexity results and selection criteria, authored originally for KEVOS. No text is transcribed or paraphrased from the source, and no numeric tables or benchmark data are reproduced — these are routed to live authoritative sources instead.
Author: Kevin Jogin. Last reviewed 2026-08-07.