ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-opus-5[1m])audited 2026-08-24

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Rational canonical form of an explicit four-by-four matrix

Example

Over $Q$ , let

$A = (\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & - 1 \\ 0 & 0 & 1 & 1 \end{array}) = C (x - 1) \oplus C ((x - 1) (x^{2} + 1)) .$

The invariant factors are $x - 1$ and $(x - 1) (x^{2} + 1)$ . Thus $A$ is already in rational canonical form,

$μ_{A} = (x - 1) (x^{2} + 1), χ_{A} = (x - 1)^{2} (x^{2} + 1) .$

Facts & Assumptions

Given: The minimal-polynomial dictionary of On a nonzero finite-dimensional space, the largest invariant factor is the minimal polynomial and the characteristic-polynomial dictionary of The product of the invariant factors is the characteristic polynomial.

[L1]

In rational canonical form, the blocks are the companion matrices of a divisibility chain of monic invariant factors (Existence and uniqueness of rational canonical form).

Verification

technique · direct

1.1

The lower three-by-three block is the companion matrix of $x^{3} - x^{2} + x - 1 = (x - 1) (x^{2} + 1)$ , while the first block is $C (x - 1) = (1)$ . Since $x - 1$ divides the cubic factor, [L1] gives the displayed polynomial-module summands and confirms that $A$ is in rational canonical form with the stated invariant factors.

2.1

The largest invariant factor in step 1.1 is $(x - 1) (x^{2} + 1)$ , so the minimal-polynomial dictionary gives the displayed $μ_{A}$ .

step 1.1given

3.1

The product of the two invariant factors is $(x - 1)^{2} (x^{2} + 1)$ , so the characteristic-polynomial dictionary gives the displayed $χ_{A}$ ; their degrees $1$ and $3$ sum to the matrix dimension.

step 1.1givenalgebra∎

Depends on

Used by

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

Dependency tree · two levels

12 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, companion-block examples (standard reference, not scraped)
A. Apisa, Wisconsin Math 542, rational canonical form examples (standard reference, not scraped)