DefinitionDefinition: Literature-sourcedProof: Not applicableprecheck passaudited 2026-08-28

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.

Self-adjoint and normal endomorphisms of a finite-dimensional real or complex inner product space

Definition

Let $V$ be a finite-dimensional real or complex inner product space and let $T : V \to V$ be an endomorphism. The endomorphism $T$ is self-adjoint when

$T^{*} = T,$

and normal when

$T T^{*} = T^{*} T,$

where $T^{*}$ is the adjoint from The adjoint $T^{*} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{*} w ⟩_{V}$ .

Every self-adjoint endomorphism is normal, because $T T^{*} = T^{2} = T^{*} T$ .

Depends on

The adjoint $T^{*} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{*} w ⟩_{V}$

Dependency tree · 2 levels

Sources

Sheldon Axler, Linear Algebra Done Right, fourth edition