TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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Existence and uniqueness of rational canonical form

Statement

Every square matrix over a field is similar to the unique rational canonical form determined by its invariant factors. If the monic invariant factors are $f_{1} ∣ \dots ∣ f_{r}$ , then

$RCF (T) = diag (C (f_{1}), \dots, C (f_{r})) .$

In rational canonical form, the blocks are the companion matrices of the invariant factors. The zero-dimensional form and invariant-factor list are empty.

Facts & Assumptions

Given: The invariant factors of an endomorphism from Invariant factors and elementary divisors of an endomorphism, the torsion statement of For finite-dimensional $V$ , $V_{T}$ is finitely generated and torsion, with annihilator generated by the minimal polynomial, the companion convention of The companion matrix of a monic polynomial, and similarity as change of basis (Similarity is an equivalence relation, and two matrices represent the same endomorphism in two bases exactly when they are similar).

[L1]

A finitely generated PID module is classified by its free rank together with its invariant factors, the latter unique up to associates in their divisibility order (Uniqueness of invariant factors and elementary divisors over a PID).

[L2]

On the power basis of a cyclic subspace, multiplication by $x$ has the companion matrix with ones on the subdiagonal and the negative coefficients in the last column (A vector annihilator gives a power basis and its companion matrix).

[L3]

Every finitely generated module over a PID $R$ is isomorphic to $R^{s} \oplus R / (a_{1}) \oplus \dots \oplus R / (a_{t})$ with each $a_{i}$ a nonzero nonunit and $a_{1} ∣ \dots ∣ a_{t}$ (Invariant-factor decomposition of a finitely generated module over a PID).

Proof

technique · constructive

1.1

Apply [L3] to the finitely generated torsion $F [x]$ -module $V_{T}$ . A free summand $F [x]^{s}$ with $s \geq 1$ contains a nonzero element with zero annihilator, so torsion forces $s = 0$ , and $V_{T} ≅ ⨁_{i = 1}^{r} F [x] / (f_{i})$ for monic nonconstant $f_{1} ∣ \dots ∣ f_{r}$ after normalizing each generator to be monic.

L3given

2.1

In the power basis of each cyclic quotient, multiplication by $x$ , hence the action of $T$ , has companion matrix $C (f_{i})$ by [L2]. Concatenating these bases gives the displayed block diagonal matrix and therefore a similarity from the original matrix to rational canonical form.

step 1.1L2construct

3.1

By [L1], the monic invariant factors are unique; unit factors give zero modules and are omitted. Thus the ordered companion-block form is unique and determines the similarity class. Dimension zero has no summands and gives the empty matrix.

step 1.1step 2.1L1discharge-construct∎

Depends on

Used by

Dependency tree · two levels

33 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

M. Brussel, Finitely Generated Modules over a PID, Theorem 7.2.1 (standard reference, not scraped)
A. Apisa, Wisconsin Math 542, Lecture 11 (standard reference, not scraped)