patronuses and animagi: a link?
My roommates and I were discussing a possible Harry Potter theory concerning patronuses and animagi. It rests on the theory that people cannot choose their animagus form. If that's false, the disregard all this, heh.

Considering that one can neither choose their animagus nor their patronis, and that their shapes are dependent on their nature, would one's animagus form be the same as their patronus?

The only case which supports this theory is McGonagall - her animagus form is a bespectacled cat, and in DH she sends out three cat patronuses. Obviously to solidify this theory we'd need more than just one example.

So far I've found a list of known animagi but no compiled list of patronuses. Theory aside, it'd be nice to have a link.

Therefore, I am wondering: are there any other instances where a character's animagus form matches their patronus? What do y'all think of this crackpot theory?

Thanks!

ETA: Spoilers for Deathly Hallows in the comments!