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 for $V$, then $\{U({\alpha }_1),\dots ,U({\alpha }_n)\}$ is also a Orthonormal Basis for $V$
5. If $B$ is any Orthonormal Basis for$V$ ,then the columns of $[U{]}_{\beta }$ are orthonormal in ${\mathbb{F}}^n$ with the standard inner product
6. If $\beta$ is any ordered Orthonormal Basis for $V$, then the rows $[U{]}_{\beta }$ are orthonormal in ${\mathbb{F}}^n$ with the standard inner product

Proofs

1. Proving Unitary Operators Property 1