====== BlockBandDiagonalize ======
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The function //BlockBandDiagonalize()// can be used to reduce the number of basis (spin-)orbitals by making linear combinations of (spin-)orbitals, according to the tight-binding structure (hopping matrix elements) within the (spin-)orbitals. As a simple example to make the idea clear, consider the following 3-by-3 matrix:
$$ M = \begin{pmatrix} 0 & 1 & 1 \\  1 & 1 & 0 \\ 1 & 0 & 1 \end{pmatrix} $$
Now, assuming that $M$ corresponds to a tight-binding Hamiltonian defined on some basis, we can linearly combine the second and third basis orbitals, such that we get a single orbital which mix with the first orbital via the matrix M. Consider the following unitary rotation matrix:
$$ U = \begin{pmatrix} 1 & 0 & 0 \\  0 & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ 0 & \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{pmatrix} $$
Now, transforming the matrix $ M $ using the unitary matrix $ U $ results in:
$$ M' = U M U^{T} = \begin{pmatrix} 0 & \frac{1}{\sqrt{2}} & 0 \\   \frac{1}{\sqrt{2}} & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} $$
In the new representation, the first basis orbital only mixes with the second 
The basis orbital and not with the third one.
The function //BlockBandDiagonalize()// can accept 3 types of objects as an arguments: [[documentation:language_reference:objects:tightbinding:start|Tight-binding object]], [[documentation:language_reference:objects:operator:start|Operator]], or [[documentation:language_reference:objects:matrix:start|Matrix]]. 
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====== Input ======
Case 1:
  * //matrix//: hermitian matrix
  * //blockSize//: size of the block (as number) or list of vectors representing the starting states

Case 2:
  * //operator//: hermitian operator
  * //wave function //: single wave function of list of wave functions

Case 3:
  * //tightbindingObject//: tight binding object
  * //startingBlock //: list of atoms with positions, shells and orbitals used as starting block


//(Optional) Third argument (in all cases)//

   *NTri : (//integer//) maximum number of blocks included (//default: $\infty$//)

   *NOrtho : (//integer//) maximum number of reorthogonalizations (//default: $\infty$//)

  *ReOrthogonalize: (//boolean//) use additional Gran-Schmidt orthogonalization after the Löwdin orthogonalization (//default: true//)

====== Output ======
   *//matrix//: block-band diagonal matrix

===== Table of contents =====
{{indexmenu>.#1}}