Unitary Matrix
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High Quality Content by WIKIPEDIA articles! In mathematics, a unitary matrix is an ntimes n complex matrix U satisfying the condition U^{dagger} U = UU^{dagger} = I_n, where In is the identity matrix in n dimensions 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 an orthogonal matrix. Just as an orthogonal matrix G preserves the (real) inner product...
High Quality Content by WIKIPEDIA articles! In mathematics, a unitary matrix is an ntimes n complex matrix U satisfying the condition U^{dagger} U = UU^{dagger} = I_n, where In is the identity matrix in n dimensions 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 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 langlecdot,cdotrangle stands now for the standard inner product on mathbb{C}^n.
