Does Aristotle's tragic hero thing apply to Hamlet?

Actually to me Hamlet IS the definition of a tragic hero. In fact most of Shakespeare's main characters are tragic heroes: Caesar Macbeth, Othello, Richard III, Romeo. My understanding of tragic hero is the main character in a tragedy. (was always taught this) Other traits common to tragic heroes are seen in Hamlet ie The hero discovers his fate by his own actions, not by things happening to him. The hero sees and understands his doom, and that his fate was revealed by his own actions..etc