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.

Read more about this topic:  Unitary Matrix

Famous quotes containing the words equivalent and/or conditions:

    “Accountability in friendship is the equivalent of love without strategy.”
    —Anita Brookner (b. 1938)

    “We have got onto slippery ice where there is no friction and so in a certain sense the conditions are ideal, but also, just because of that, we are unable to walk. We want to walk so we need friction. Back to the rough ground!”
    —Ludwig Wittgenstein (1889–1951)