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TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16
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  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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The restriction of a diagonalisable endomorphism to an invariant subspace is diagonalisable

Statement

If T:V→V is diagonalisable and W⊆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 μT (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
1.1L1L2L3L4algebra

By [L2], μT is split and squarefree. By [L1], μT∣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.

2.1step 1.1L2∎

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

Depends on

Dependency tree · 2 levels

Sources