Aristotle said "A man doesn't become a hero until he can see the root of his own downfall." A tragic hero exercises flawed judgment (hamartia), caused by excessive pride or hubris, which causes his own downfall. The hero also realizes his downfall was his own doing.
In Julius Caesar, Brutus is a tragic hero. His decision to kill Caesar is flawed and caused by excessive pride because he believes he knows what is best for Rome. Brutus loves Caesar, but when Cassius asks him if he feels content with Caesar's growing acclaim in Act I, Scene 2, Brutus answers, "I would not, Cassius; yet I love him well." Brutus loves Caesar and would rather not kill him, but he allows Cassius to manipulate him into killing Caesar. In Act V, Scene 5, just before he kills himself by running through his sword, Brutus says, "Caesar, now be still: I kill'd not thee with half so good a will." His words show an awareness of the flawed judgement that led to his own downfall, and this awareness is another feature of a tragic hero. In the end, Brutus feels remorseful about having thought, with excessive pride, that his actions could save Rome.