"The" Eagles are officially just Eagles. The "the" isn't part of their name. From their wiki:
Steve Martin, a friend of the band from their early days at The Troubadour, recounts in his autobiography that he suggested that they should be referred to as "the Eagles", but Frey insists that the group's name is simply "Eagles".
Steve Martin, a friend of the band from their early days at The Troubadour, recounts in his autobiography that he suggested that they should be referred to as "the Eagles", but Frey insists that the group's name is simply "Eagles".