Self-adjoint 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 self-adjoint when $T^* = T$ , where $T^*$ is the adjoint of $T$ . Equivalently, $\langle T(v), w \rangle = \langle v, T(w) \rangle$ for all $v, w \in V$ .

Remarks

In an orthonormal basis, $T$ is self-adjoint if and only if its matrix is symmetric (over $\R$ ) or Hermitian (over $\C$ ). Orthogonal projections are self-adjoint. The eigenvalues of a self-adjoint map are real, see the reality of the eigenvalues. Self-adjoint maps are the subject of the spectral theorem.

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
Self-adjoint operator definition

Where this leads

19 pages build on this one.

Tag	`0MB8HQW495`
URI	`m2://0MB8HQW495`
Kind	definition
Writing	W1 stated
MSC	47B15, 15B57
Defines	Self-adjoint operator
Uses	Adjoint of a linear map, Endomorphism

Self-adjoint operator

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

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