I'm currently writing my dissertation and I'm stuck. I need to solve the non linear matrix equation
$$ \mathbf{FC}=\mathbf{C}\boldsymbol{\varepsilon} $$
where the matrix $ \mathbf{F} $ depends the elements of the matrix $\mathbf{C}$;
the diagonalization procedure is iterative.
The matrix $\mathbf{C^{-1}FC}$ is symmetric and is partitioned as follow:
Once the matrix $ \mathbf{F} $ is diagonalized, the total energy can be calculated as the sum of the occupied orbital diagonal elements of the matrix $ \boldsymbol{\varepsilon} $ plus other various terms:
$$ \mathcal{E}={\displaystyle \sum_{\mathrm{occ}}}\varepsilon+\mathrm{other\, terms} $$ Here's the point: I don't understand why, given the fact that the energy is invariant with respect to unitary transformations among occupied orbitals and among virtual orbitals, the full iterative diagonalization can be replaced by a pseudodigonalization procedure which consists in the annihilation of virtual-occupied blocks of the matrix $\mathbf{C^{-1}FC}$.
Thanks in advance!