Vertigo's villain, Elster, is a flat character who makes only three on-screen appearances:
during his first meeting with Scottie,
at a follow-up rendezvous to discuss Madeleine with the detective, and
at the trial where Madeleine's death is declared a suicide.
All we know about the guy is that he's an old college buddy of Scottie's who's made a fortune in the ship-building biz. He's a little stiff and condescending, but at first he's genuinely concerned about his wife, who's been acting very strange and who he thinks might be possessed by the spirit of one of her ancestors. He pressures Scottie to investigate, even though Scottie tells him he's retired. What a good husband. He plays the bereaved hubby after "Madeleine's" death and reassures Scottie that Scottie shouldn't feel at all responsible.
We learn two-thirds of the way through the film that Elster's truly diabolical. He hatched a plan to murder his wife and hired Judy to help make it happen by impersonating his wife. It only becomes clear just how evil Elster is in retrospect, late in Vertigo's game. He's also clever; he chose Scottie because of his fear of heights, knowing he couldn't climb the tower steps in time to save "Madeleine." By then, he's long gone. He seems to have gotten away with it. Scottie learns from Judy that Elster killed the real Madeleine, but it's clear that even if he doesn't jump off the bell tower himself, he's not going to be in any shape to do anything about it.
See our "Mode of Production" section for some fun facts about the alternate ending that gives Gavin Elster his just deserts.