Can every diagonalizable matrix be diagonalized into the identity matrix? I'm a chemistry major and I haven't taken much math, but this came up in a discussion of quantum chemistry and my professor said (not very confidently) that if a matrix is diagonalizable, then you should be able to diagonalize it to the identity matrix. I suspect this is true for symmetrical matrices, but not all matrices. Is that correct?

The usual meaning of "diagonalization" is diagonalization by similarity transform, which takes the form of $PAP^{-1}=D$. As the others have shown, the only matrix that can be diagonalized into the identity matrix is the identity matrix itself.
Yet, depending on the context, your professor may refer to diagonalization by congruence, which takes the form of $P^\ast AP=D$. If he implicitly assumes that $A$ is positive definite, then his assertion is true: $P^\ast AP=D\,\Rightarrow\,(PD^{-1/2})^\ast A(PD^{-1/2})=I$. However, since I know absolutely nothing about quantum chemistry, I can't say if he really meant this.
