Normal 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 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.

Explanations

L1 human

A normal map commutes with its adjoint. This condition is weaker than self-adjoint, but it still gives perpendicular eigenvectors over $\C$. Rotations of the plane are normal, but not self-adjoint.