Update: We've got it under control! When I took him for his rabies shot, I asked the vet what she recommends, and she told me about this:

http://www.furminatorcat.com/

I had to think twice about buying it because it was really expensive, but since getting it, he hasn't had a mat. It's a little scary reading the warnings that come with it, but luckily he loves it, and is reacting well to it (I read reviews, and not all cats do). I haven't had as much fur on my clothes or rugs either. I'm still brushing him daily with his regular brush since the furminator doesn't remove existing mats, but it's really made a huge difference in preventing new ones.