Unitary matrices

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A unitary matrix is a complex matrix U satisfying the condition

U^\dagger U = UU^\dagger = I_n\,

where I is the identity matrix and U^\dagger 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^\dagger \,

U^{-1} = U^\dagger.

A unitary matrix in which all entries are real is called an orthogonal matrix.