Linear algebra MOC

Unitary operator

A unitary operator is a bounded linear operator on a Hilbert space which preserves the inner product, #m/def/linalg i.e. for any

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

Properties

Let be a unitary operator

• The Spectrum of lies on the unit circle, i.e. each eigenvalue has .

Matrix

Let be a unitary matrix. Then

• Both the columns and rows of form an Orthonormal basis with respect to the inner product.
• is an isometry with respect to the 2-norm, i.e. for all .

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