Unitary Matrix

linear-algebra

Definition

Unitary Matrix

Let $U$ be a square matrix over the complex numbers. The matrix is unitary if its conjugate transpose is also its inverse, that is,

$$U^{\ast }U=U{U}^{\ast }=I$$

where $I$ is the identity matrix. Equivalently, $U$ preserves the standard inner product:

$$⟨Ux,Uy⟩=⟨x,y⟩\forall x,y\in {\mathbb{C}}^n$$

Properties

Isometric

Unitary matrices are isometric, i.e.:

$$||Ux||=||x||\forall x\in {\mathbb{C}}^n$$

Eigenvalues

Eigenvalues of a unitary matrix are unimodular, i.e.:

$$|\lambda |=1\forall \lambda \in E_{\lambda }(U)$$