ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-24

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One matrix in rational form over $Q$ and Jordan form after adjoining its roots

Example

For the preceding matrix $A$ , rational canonical form over $Q$ has companion blocks $C (x - 1)$ and $C ((x - 1) (x^{2} + 1))$ . Over a field containing $i$ with $i^{2} = - 1$ , its Jordan form is

$diag (1, 1, i, - i) .$

Facts & Assumptions

Given: Jordan form from module elementary divisors (Jordan canonical form from the elementary divisors of $V_{T}$ ) and agreement with Jordan-string block data (For split characteristic polynomial, elementary divisors and Jordan strings give the same Jordan blocks).

[L1]

The invariant factors are $x - 1$ and $(x - 1) (x^{2} + 1)$ (Rational canonical form of an explicit four-by-four matrix).

Verification

technique · direct

1.1

Over $Q$ , [L1] gives the two rational companion blocks and the divisibility chain, so no splitting field is needed for rational canonical form.

2.1

After adjoining $i$ , the second invariant factor splits as $(x - 1) (x - i) (x + i)$ into pairwise coprime factors, while the first contributes another $x - 1$ . The elementary divisors are therefore $x - 1, x - 1, x - i, x + i$ .

step 1.1algebra

3.1

Each elementary divisor in step 2.1 is linear, so the module Jordan theorem gives four one-dimensional blocks with entries $1, 1, i, - i$ , and the agreement theorem confirms the same Jordan-string block data. Their product remains $(x - 1)^{2} (x^{2} + 1)$ and their least common multiple remains $(x - 1) (x^{2} + 1)$ , preserving the characteristic and minimal polynomial identities.

step 2.1given∎

Depends on

Used by

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Dependency tree · two levels

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Sources

M. Brussel, Finitely Generated Modules over a PID, canonical-form comparison (standard reference, not scraped)
A. Apisa, Wisconsin Math 542, Lectures 11-12 (standard reference, not scraped)

One matrix in rational form over QQ and Jordan form after adjoining its roots