CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck 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

1.1L1algebra

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.

2.1step 1.1∎

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.

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 · two levels

8 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

Keith Conrad, The Minimal Polynomial and Some Applications, Corollary 5.5 (standard reference, not scraped)