Most linear algebra discussions start by assuming the matrices are expressed in an orthonormal basis. Sometimes its computationally favorable to work with nonorthogonal bases because the matrix will be sparse.

In electronic structure, using a basis of localized atomic-like functions is good for systems with localized d electrons, for example. This basis consists of s, p, d, f, etc. states on each atom in the crystal. Clearly basis elements on different atoms will be nonorthogonal.

This requires most of the identities and equations to be generalized a bit. These notes discuss what happens.