---
tag: 0MB8HQW495
kind: definition
name: Self-adjoint operator
msc: [47B15, 15B57]
defines: 0NEGN5CRSC
uses: [0ST8QF9T3T, 0PTF1PTJK0]
sources:
  - url: https://en.wikipedia.org/wiki/Self-adjoint_operator
    locator: Definitions
---

## Definition

Let $V$ be a finite-dimensional inner product space over $\R$ or $\C$, and let $T \colon V \to V$ be an [[endomorphism|0J7H0YBFDJ]].
The map $T$ is *self-adjoint* when $T^* = T$, where $T^*$ is the [[adjoint|0R9NMSM1ZP]] 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|0VR671PHE5]] (over $\R$) or [[Hermitian|0P17NGQQMQ]] (over $\C$).
Orthogonal projections are self-adjoint.
The eigenvalues of a self-adjoint map are real, see [[the reality of the eigenvalues|0C4NT4GT3X]].
Self-adjoint maps are the subject of the [[spectral theorem|06J4PXRG22]].

## Explanations

### L1 human

A self-adjoint map can move freely from one side of the inner product to the other.
For real matrices, these are the symmetric matrices.
They model observable quantities in quantum mechanics, because their eigenvalues are real.