Hello Math Stackexchange community,
I am trying to numerically solve the Schrodinger equation for a 1-particle Hamiltonian using a finite non-orthogonal basis (see the attached image), HC=SCE. My approach is to first transform the initial non-orthogonal basis into a new orthogonal basis. In the matrix form, this leads to the transformation matrix X (see the attached image), and I want to know the properties of this matrix. If my derivations are correct, X is a unitary matrix, which does not make sense to me because it appears that the initial (non-orthogonal) basis set must be orthogonal. Could you please point out if there is anything wrong with my reasoning? I can comment more on my question or reasoning if they are unclear. Assume that X is invertible.
