Unitary Operator

linear-algebra

Definition

Unitary Operator

A unitary operator on a complex inner product space is a linear operator $U$ satisfying

$$U^{†}U=U{U}^{†}=I,$$

where $U^{†}$ denotes the adjoint of $U$ and $I$ is the identity operator. Equivalently, a unitary operator preserves inner products, that is,

$$⟨Ux,Uy⟩=⟨x,y⟩$$

for all vectors $x$ and $y$. In particular, unitary operators preserve norms and are therefore the basic form of time evolution for closed quantum systems.

Example: The matrix

$$U=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$$

is unitary, since it is real and orthogonal, and hence satisfies $U^{†}U=I$.