Unitary matrix

In mathematics, a unitary matrix is an  complex matrix U satisfying the condition
where In is the identity matrix in n dimensions and  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
A unitary matrix in which all entries are real is an orthogonal matrix. Just as an orthogonal matrix G preserves the (realinner productof two real vectors,
so also a unitary matrix U satisfies
for all complex vectors x and y, where  stands now for the standard inner product on .
If  is an n by n matrix then the following are all equivalent conditions:
1.  is unitary
2.  is unitary
3. the columns of  form an orthonormal basis of  with respect to this inner product
4. the rows of  form an orthonormal basis of  with respect to this inner product
5.  is an isometry with respect to the norm from this inner product
6.  is a normal matrix with eigenvalues lying on the unit circle.

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