Unitary Operator

Definition

A linear operator $U$ on an inner product space is unitary if it preserves inner products:

$$ \langle Ux, Uy \rangle = \langle x, y \rangle \quad \text{for all } x, y $$

Equivalently, $U^* U = U U^* = I$, where $U^*$ is the adjoint (conjugate transpose).

For Matrices

A complex matrix $U$ is unitary if:

$$ U^\dagger U = U U^\dagger = I $$

where $U^\dagger = \overline{U^T}$ is the conjugate transpose.

For real matrices, this simplifies to orthogonal: $U^T U = I$.

Properties

Unitary operators preserve:

Property	Formula
Inner products	$\langle Ux, Uy \rangle = \langle x, y \rangle$
Norms	$\|Ux\| = \|x\|$
Angles	The angle between vectors is unchanged
Orthogonality	Orthogonal vectors remain orthogonal

Intuition

Unitary operators are the “rigid motions” of inner product spaces:

• Rotations are unitary
• Reflections are unitary
• Stretching is NOT unitary (changes lengths)

They rearrange vectors without distorting the geometry.

Examples

• Rotation matrices in $\mathbb{R}^n$
• The Fourier transform on $L^2(\mathbb{R})$ (Plancherel theorem)
• Quantum mechanical time evolution operators

Related Concepts

• Inner product — what unitary operators preserve
• Norm — also preserved by unitary operators
• Isomorphism — unitary operators are isomorphisms of Hilbert spaces