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.

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