Unitary Matrix - Equivalent Conditions

Equivalent Conditions

If U is a square, complex matrix, then the following conditions are equivalent:

1. U is unitary
2. U * is unitary
3. U is invertible, with U –1=U *.
4. the columns of U form an orthonormal basis of with respect to the usual inner product
5. the rows of U form an orthonormal basis of with respect to the usual inner product
6. U is an isometry with respect to the usual norm
7. U is a normal matrix with eigenvalues lying on the unit circle.