DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Jordan bases and Jordan canonical forms over the base field

Definition

A Jordan basis over $F$ for an endomorphism $T : V \to V$ is an ordered basis obtained by concatenating Jordan strings for $T$ , all with eigenvalues in $F$ (Jordan blocks, Jordan strings, and their endpoints). The matrix of $T$ in such a basis is block diagonal with blocks $J_{m} (λ)$ and is called a Jordan canonical form of $T$ over $F$ .

The adjective “canonical” records the block multiset, not a prescribed order of blocks: permuting the strings usually changes the literal block diagonal matrix. On the zero space the empty basis and empty block matrix form the Jordan basis and Jordan form.

Depends on

Jordan blocks, Jordan strings, and their endpoints

Used by

Dependency tree · two levels

3 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

S. Axler, Linear Algebra Done Right, 4th ed., Section 8C (standard reference, not scraped)
S. Treil, Linear Algebra Done Wrong, Chapter 9, Sections 4-5 (standard reference, not scraped)