Linear algebra MOC

Unitary operator

A unitary operator is a bounded linear operator 𝑈 :𝐻 →𝐻 on a Hilbert space 𝐻 which preserves the inner product, linalg i.e. for any 𝑣,𝑤 ∈𝐻

⟨𝑈𝑣|𝑈𝑤⟩=⟨𝑣|𝑤⟩

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

𝑈†𝑈=𝑈𝑈†=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. ∥𝐔⃗𝐯∥2 =‖⃗𝐯‖2 for all ⃗𝐯 ∈ℂ𝑛.

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