Unitary matrix

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 it has an inverse which is equal to its conjugate transpose $U^* \,$ .

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 <.,.> stands now for the standard inner product on Cⁿ. If

A \,

is an n by n matrix then the following are all equivalent conditions:

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

where V is unitary, and $\Sigma$ is diagonal and unitary.

For any n, the set of all n by n unitary matrices with matrix multiplication form a group.

A unitary matrix is called special if its determinant is 1.

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See also