As one of my short friends put it, "If you're short, you gotta be tough." Short guy syndrome describes the tendency of short males to engage in macho activities in order to counteract the stereotypes of short men. In the case of American males who are short (say, 5'5" and under), the stereotypes suggest "weak" and "wimpy" as keywords. This leads to a desire to go to the gym and build muscle tone, play extreme sports, buy guns (to shoot bullets, paintballs, etc.), and engage in other activities to show the rest of the world that you don't take shit from anyone.
Short guy syndrome is not linked with any sort of outward hostility towards other people. I don't think a short person has ever killed a tall person out of jealousy or resentment for height.