Self-adjoint operator

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.

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.