Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 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.

Every finite-dimensional endomorphism over an algebraically closed field has Jordan form

Statement

Every endomorphism of a finite-dimensional vector space over an algebraically closed field has a Jordan canonical form over that field, including the endomorphism of the zero space.

Facts & Assumptions

Given: A finite-dimensional vector space over an algebraically closed field F and an endomorphism T.

[L1]

Every nonconstant polynomial over an algebraically closed field has a root in that field (An algebraically closed field: every nonconstant polynomial has a root in the field).

[L2]

If a is a root of p∈F[x], then p=(x−a)q for some q∈F[x] (Factor theorem over a commutative ring).

[L3]

An endomorphism has Jordan form over F exactly when its characteristic polynomial splits over F (Jordan form over the base field exists exactly when the characteristic polynomial splits).

Proof

technique · induction on $\deg\chi_T$
1.1baseL3

If deg⁡χT=0, then χT=1 and [L3] gives the empty Jordan form.

1.2L1L2ih

The induction is on the degree of an arbitrary nonzero polynomial over F, not only of a characteristic polynomial, since the factor produced below need not itself be one. If the degree is positive, [L1] supplies a root a∈F and [L2] writes χT=(x−a)q with deg⁡q smaller; the induction hypothesis applies to q in that strengthened form and factors it into linear factors, so χT splits.

2.1step 1.1step 1.2L3discharge-induction∎

Fact [L3] now gives a Jordan canonical form for T, completing the induction.

Depends on

Used by

Dependency tree · two levels

12 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