DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-13

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.

Algebraic multiplicity as the exponent of $x - λ$ in $χ_{T}$ , and geometric multiplicity as $\dim E_{λ} (T)$

Definition

Let $T : V \to V$ be an endomorphism of a finite-dimensional vector space over $F$ , and let $λ \in σ_{F} (T)$ . The algebraic multiplicity of $λ$ is the largest natural number $m \geq 1$ such that $(x - λ)^{m}$ divides $χ_{T} (x)$ in $F [x]$ . This largest exponent exists because $λ$ is a root, the factor theorem gives one factor $x - λ$ , and the nonzero polynomial $χ_{T}$ has finite degree and unique factorization.

The geometric multiplicity of $λ$ is

$\dim_{F} E_{λ} (T) .$

The eigenspace is a nonzero subspace of the finite-dimensional space $V$ , so this dimension is defined and is at least $1$ .

Algebraic multiplicity as the exponent of $x - λ$ in $χ_{T}$ , and geometric multiplicity as $\dim E_{λ} (T)$

Definition

Depends on

Dependency tree · 2 levels

Sources

Algebraic multiplicity as the exponent of x−λx−λ in χTχT​, and geometric multiplicity as dim⁡Eλ(T)dimEλ​(T)

Definition

Depends on

Dependency tree · 2 levels

Sources

Algebraic multiplicity as the exponent of $x - λ$ in $χ_{T}$ , and geometric multiplicity as $\dim E_{λ} (T)$