ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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The identity and a nontrivial Jordan block have the same characteristic polynomial but different minimal polynomials

Example

Over any field, let

$I = (\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}), J = (\begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array}) .$

Both matrices have characteristic polynomial $(x - 1)^{2}$ , but $μ_{I} = x - 1$ while $μ_{J} = (x - 1)^{2}$ .

Facts & Assumptions

Given: The matrices $I, J$ in the Example.

[L1]

A block-triangular characteristic polynomial is the product of the characteristic polynomials of its diagonal blocks (The characteristic polynomial of a block upper- or lower-triangular matrix is the product of the characteristic polynomials of its diagonal blocks).

[L2]

The minimal polynomial is the monic polynomial of least degree that annihilates the matrix (The annihilator ideal is nonzero and has a unique monic generator; $p (T) = 0$ if and only if $μ_{T} ∣ p$ ).

Verification

technique · direct

1.1

By [L1], both characteristic polynomials are $(x - 1)^{2}$ .

2.1

The polynomial $x - 1$ annihilates $I$ . For $N = J - I \neq 0$ , one has $N^{2} = 0$ , so $(J - I)^{2} = 0$ but $J - I \neq 0$ . Hence [L2] gives $μ_{I} = x - 1$ and $μ_{J} = (x - 1)^{2}$ .

L2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 40 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Keith Conrad, The Minimal Polynomial and Some Applications, Example 4.2 (standard reference, not scraped)