TheoremStatement: Literature-sourcedProof: 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 restriction of a diagonalisable endomorphism to an invariant subspace is diagonalisable

Statement

If $T : V \to V$ is diagonalisable and $W \subseteq V$ is $T$ -invariant, then the restriction $T ∣_{W}$ is diagonalisable.

Facts & Assumptions

Given: A diagonalisable endomorphism $T$ and a $T$ -invariant subspace $W$ .

[L1]

The minimal polynomial of $T ∣_{W}$ divides $μ_{T}$ (The minimal polynomial of a restriction to an invariant subspace divides the original minimal polynomial).

[L2]

Diagonalisability is equivalent to the minimal polynomial being a product of distinct linear factors (An endomorphism is diagonalisable if and only if its minimal polynomial is a product of distinct linear factors).

[L3]

A polynomial splits over $F$ when it factors into linear factors over $F$ (Polynomials that split and splitting fields of a polynomial or a family of polynomials).

[L4]

The polynomial ring over a field is a unique factorisation domain (For every field $F$ , $F [x]$ is a unique factorisation domain).

Proof

technique · direct

1.1

By [L2], $μ_{T}$ is split and squarefree. By [L1], $μ_{T ∣_{W}}$ is a monic divisor of it, so unique factorisation [L4] and the meaning of splitting in [L3] make the restriction polynomial split and squarefree as well.

L1L2L3L4algebra

2.1

Apply the reverse implication of [L2] to $T ∣_{W}$ . This includes $W = 0$ , whose minimal polynomial is $1$ .

step 1.1L2∎

Depends on

Used by

A family of diagonalisable endomorphisms of a finite-dimensional space is simultaneously diagonalisable if and only if its members commute pairwise Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 44 results over 7 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, §5, Lemma 5.1 (standard reference, not scraped)