DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28

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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 $\langle Tv,w\rangle_W=\langle v,T^*w\rangle_V$

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

Sheldon Axler, Linear Algebra Done Right, fourth edition (standard reference, not scraped)