Unitary matrix

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In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition

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

where $I_n\,$ is the identity matrix and $U^* \,$ is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if and only if it has an inverse which is equal to its conjugate transpose $U^* \,$

$U^{-1} = U^* \,\;$

A unitary matrix in which all entries are real is the same thing as an orthogonal matrix. Just as an orthogonal matrix G preserves the (real) inner product of two real vectors,

$\langle Gx, Gy \rangle = \langle x, y \rangle$

so also a unitary matrix U satisfies

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

for all complex vectors x and y, where $\langle.,.\rangle$ stands now for the standard inner product on Cⁿ. If $U \,$ is an n by n matrix then the following are all equivalent conditions:

$U \,$ is unitary
$U^* \,$ is unitary
the columns of $U \,$ form an orthonormal basis of Cⁿ with respect to this inner product
the rows of $U \,$ form an orthonormal basis of Cⁿ with respect to this inner product
$U \,$ is an isometry with respect to the norm from this inner product

It follows from the isometry property that all eigenvalues of a unitary matrix are complex numbers of absolute value 1 (i.e. they lie on the unit circle centered at 0 in the complex plane). The same is true for the determinant.

All unitary matrices are normal, and the spectral theorem therefore applies to them. Thus every unitary matrix U has a decomposition of the form

$U = V\Sigma V^*\;$