Normal operator

Builds on Adjoint of a linear map Endomorphism

Definition

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$ , and let $T \colon V \to V$ be an endomorphism. The map $T$ is normal when $T T^* = T^* T,$ where $T^*$ is the adjoint of $T$ .

Remarks

Self-adjoint maps, skew-adjoint maps and unitary maps are normal. $T$ is normal if and only if $\norm{T(v)} = \norm{T^*(v)}$ for all $v \in V$ . Over $\C$ , normal maps are exactly the maps with an orthonormal basis of eigenvectors.

Reading path

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Vector space definition
Inner product definition
Linear map definition
Inner product space definition
Endomorphism definition
Adjoint of a linear map definition
Normal operator definition

Tag	`0ENZN5G4CV`
URI	`m2://0ENZN5G4CV`
Kind	definition
Writing	W1 stated
MSC	47B15, 15A04
Defines	Normal operator
Uses	Adjoint of a linear map, Endomorphism

Normal operator

Definition

Remarks

Reading path

Further reading

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