Unitary operator

A unitary operator is a bounded linear operator $𝑈 : 𝐻 \to 𝐻$ on a Hilbert space $𝐻$ which preserves the inner product, #m/def/linalg i.e. for any $𝑣, 𝑤 \in 𝐻$

⟨ 𝑈 𝑣 | 𝑈 𝑤 ⟩ = ⟨ 𝑣 | 𝑤 ⟩

An equivalent condition is that the Hermitian conjugate of $𝑈$ is its inverse, i.e.

𝑈 † 𝑈 = 𝑈 𝑈 † = 𝐼

Properties

Let $𝑈$ be a unitary operator

The Spectrum of $𝑈$ lies on the unit circle, i.e. each eigenvalue $𝜆$ has $| 𝜆 | = 1$ .

Matrix

Let $𝐔$ be a unitary matrix. Then

Both the columns and rows of $𝐔$ form an Orthonormal basis with respect to the inner product.
$𝐔$ is an isometry with respect to the 2-norm, i.e. $∥ 𝐔 ⃗ 𝐯 ∥ 2 = ‖ ⃗ 𝐯 ‖ 2$ for all $⃗ 𝐯 \in ℂ 𝑛$ .

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