Unitary matrix

$U^* U = UU^* = I \,$

where I is the identity matrix and U * is the conjugate transpose of U.

The real analogue of a unitary matrix is an orthogonal matrix.

Properties [edit]

For any unitary matrix U, the following hold:

Given two complex vectors x and y, multiplication by U preserves their inner product; that is,

$\langle Ux, Uy \rangle = \langle x, y \rangle$ .

U is normal
U is diagonalizable; that is, U is unitarily similar to a diagonal matrix, as a consequence of the spectral theorem. Thus U has a decomposition of the form

$U = VDV^*\;$

where V is unitary and D is diagonal and unitary.

$|\det(U)|=1$ .
Its eigenspaces are orthogonal.
For any positive integer n, the set of all n by n unitary matrices with matrix multiplication forms a group, called the unitary group U(n).
Any square matrix with unit Euclidean norm is the average of two unitary matrices.^[1]

If U is a square, complex matrix, then the following conditions are equivalent:

U is unitary
U * is unitary
U is invertible, with U ^–1=U *.
the columns of U form an orthonormal basis of $\mathbb{C}^n$ with respect to the usual inner product
the rows of U form an orthonormal basis of $\mathbb{C}^n$ with respect to the usual inner product
U is an isometry with respect to the usual norm
U is a normal matrix with eigenvalues lying on the unit circle.

^ Li, Chi-Kwong; Poon, Edward (2002). "Additive Decomposition of Real Matrices". Linear and Multilinear Algebra 50 (4): 321–326. doi:10.1080/03081080290025507.