CorollaryStatement: 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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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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If two commuting endomorphisms are diagonalisable, then every finite linear combination of products of their powers is diagonalisable; in particular, their sum and product are diagonalisable

Statement

Let $S, T$ be commuting diagonalisable endomorphisms. Every endomorphism of the form

$\sum_{j < m} c_{j} S^{a_{j}} T^{b_{j}}$

with $m, a_{j}, b_{j} \in N$ and $c_{j} \in F$ is diagonalisable. In particular, $S + T$ and $S T$ are diagonalisable.

Facts & Assumptions

Given: Commuting diagonalisable endomorphisms $S, T$ and the displayed finite polynomial expression.

[L1]

A pairwise commuting family of diagonalisable endomorphisms has a common eigenbasis (A family of diagonalisable endomorphisms of a finite-dimensional space is simultaneously diagonalisable if and only if its members commute pairwise).

Proof

technique · direct

1.1

By [L1], choose a basis in which both $S$ and $T$ are diagonal. Every power, product of powers, and finite linear combination in the Statement remains diagonal in that same basis.

L1algebra

2.1

Thus every displayed expression is diagonalisable. Taking the expressions $S + T$ and $S T$ gives the stated special cases; the empty sum and zero space cause no exception.

step 1.1∎

Depends on

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

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: 32 results over 14 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, Corollary 5.5 (standard reference, not scraped)