Yes, I believe so, up to some multiplicative factor.

You can carry out the exponentiations in the field Q[sqrt(5)], which is two-dimentional over Q. The interesting thing here is that diagonalization is trading one 2d vector space for another -- one with a very well-behaved multiplication operation (it's distributive, commutative, associative, and invertible).

There's no need to do this to quickly compute fibonacci numbers, but it's pretty neat.