Is the Lowdin Orthogonalization used in diagonalizing the atomic orbitals really a basis change? This is kind of more on the mathematics side of quantum chemistry, but I can't quite figure out why the Lowdin Orthogonalization is called a basis set change. I get how it works from the perspective of matrices, and that it creates an orthonormal set as long as you have linearly independent input, but I don't get why it works as a change of basis.

Typically to change the basis set of a matrix $M$ you use a non-singular linear transformation $A$, which need not be unitary, as such: $$M'=A^{-1}MA.$$ Quantum mechanics, we are dealing with Hilbert Spaces, which produce an isometry between the Hilbert space and the dual (I think?), and then for post HF methods most methods often use unitary transformations, so that $A^\dagger=A^{-1}$, and so it becomes trivial. 

The Lowdin transformation between nonorthogonal AO $|\phi\rangle$ and orthogonal AO $|\phi\rangle_\perp$ involves the overlap matrix $S$ as: 
$$ |\phi \rangle_\perp = S^{-\frac{1}{2}}|\phi\rangle  $$ 
Then, the corresponding bra just has the adjoint, which is also $S^{\frac{-1}{2}}$ because the matrix is self-adjoint: $$ \langle \phi|_\perp = \langle \phi|(S^{-\frac{1}{2}})^\dagger = \langle \phi|S^{-\frac{1}{2}}$$

But, considering the matrix analog, why don't we use the inverse? 

For instance, we have the identity: $$ D^{AO} = C  D^{MO}C^T $$ where $D$ is the 1-electron reduced density matrix. In my thinking this has to be a transformation from the MO to the AO on the right of $D^{MO}$, and from the AO to the MO on the left, which gives you $D^{AO}$. However, $C$ is not unitary obviously, because it comes from the Lowdin procedure, and so while $C$ on the left is from the AO to the MO, $C^{-1}$ is not equivalent to $C^T$. 

Furthermore to get $D^{MO}$ matrix in terms of the $D^{AO}$ you DO have to apply the inverse relationship with $C^{-1}$. $$D^{MO} = C^{-1}D^{AO}(C^T)^{-1},$$ because $C^{-1}$ and $C$ and related non-trivially through $S$. 

Hopefully this is appropriate to ask here. I think the answer involves Hilbert spaces but I can't find a lot on non-orthogonal transformations. Thanks!
Consider a nonsingular, linear transformation $\mathbf A$ of a set of vectors arranged into a matrix $\mathbf V$ $$\mathbf V'=\mathbf {VA}$$

We can say the matrix $\mathbf V'$ is orthogonal (and thus that we have linearly transformed the vectors to form an orthogonal basis) if $$\mathbf {V'^{\dagger}V'}=\mathbf {A^{\dagger}V^{\dagger}VA}=\mathbf {A^{\dagger}SA}=\mathbf{1}$$ 

In general, this occurs when $\mathbf{A}=\mathbf{S}^{-1/2}\mathbf{B}$, where $\mathbf{B}$ can be any unitary matrix. The Lowdin orthogonalization commonly used is just the case where $\mathbf{B=1}$, so we can see that we are properly changing the basis. 

But your issue is specifically with transformations of the density matrix, where the transformation matrix is not unitary in general. The standard change of basis formalism would suggest that to go to AO basis from MO, we would do a transformation something like this:
$$\mathbf{C}\mathbf{D}^{MO}\mathbf{C}^{-1}=\mathbf{D}^{AO}\mathbf{S}$$
Ensuring orthogonality of the MOs requires that $\mathbf{C^{\dagger}\mathbf{S}\mathbf{C}}=\mathbf{1}$, which we can sub in to the equation above to give: $$\mathbf{C}\mathbf{D}^{MO}\mathbf{C^{\dagger}\mathbf{S}\mathbf{C}}\mathbf{C}^{-1}=\mathbf{C}\mathbf{D}^{MO}\mathbf{C}^{\dagger}\mathbf{S} =\mathbf{D}^{AO}\mathbf{S} \to \mathbf{C}\mathbf{D}^{MO}\mathbf{C}^{\dagger}=\mathbf{D}^{AO}$$

So the form in terms $\mathbf C$ and its adjoint is a direct result of the conventional similarity transform combined with the orthonormality condition on the MOs. 

This latter derivation was adapted from$^1$ which can be found here.

As to why we avoid using transformations involving the inverse of the transformation matrix, I suspect Zhe is correct that it is a matter of cost, since obtaining the transpose/adjoint of a matrix requires almost no work at all, while an inverse is relatively difficult to obtain.


T. Helgaker et. al Chemical Physics Letters 327 (2000). 397–403