Unitary Operator

Definition

A unitary operator on an Inner Product Space is an (inner product) vector isomorphism of the space to itself.

Formal Definition

$T\in \mathcal{L}(V)$ s.t $⟨T\alpha |T\beta ⟩=⟨\alpha |\beta ⟩,\forall \alpha ,\beta \in V$ Its a isomorphism that Preserves Inner Products

Equivalent Definitions

1. $U{U}^{\ast }=I=U^{\ast }U$
2. $U$ is Preserving Inner Products and Vector Isomorphism ($⟨U\alpha ,U\beta ⟩=⟨\alpha |\beta ⟩$)
3. $U$ is Norm preserving. ($||U\alpha ||=||\alpha ||,\alpha \in V$)
4. if $\{{\alpha }_1,\dots ,{\alpha }_n\}$ is an Orthnormal Basis, then $$