Unitary Operator

linear-algebra

Definition

Unitary Operator

Let $U:\mathcal{H}\to \mathcal{H}$ be a linear operator on a complex inner product space $\mathcal{H}$. The operator is unitary if its adjoint $U^{†}$ is also its inverse, that is,

$$U^{†}U=U{U}^{†}=I$$

where $I$ is the identity operator. Equivalently, $U$ preserves inner products:

$$⟨Ux,Uy⟩=⟨x,y⟩\forall x,y\in \mathcal{H}$$

Properties

Isometric

Unitary operators $T:V\to V$ are isometric, i.e.:

$$||T_x||=||x||\forall x\in V$$

Eigenvalues

Eigenvalues of a unitary operator $T$ are unimodular, i.e. it holds that:

$$|\lambda |=1\forall \lambda \in E_{\lambda }(T)$$