Elimination of the Overlap Matrix

Equations (1.17) are solved by first eliminating matrix S(k). This is carried out most simply with the aid of Lowdin s symmetric orthogonali-zation procedure. If equations (1.17) are multiplied on the left by S(k) and S(k) S(k) = 1 inserted, we obtain 

1 If 00, or is finite but large, the individual energy levels within a band are closer than ktT (k is the Boltzmann constant) and therefore the level distribution within a band can be r arded as continuous. 

The square root and inverse square root of S(k) are easily calculated via its diagonalization, namely 

The simplest way to diagonalize the Hermitian complex matrix S(k) defined by equation (1.19) is to rewrite its eigenvalue equation (1.25) in the form 

The latter two matrix equations can be written as the single matrix eigenvalue equation 

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