Appendix E 
INVERSION OF LARGE BAND MATRICES 
("Piggy up a hill" 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
Yeo The method of orthogonalisation of a matrix T consists 
in premultiplying T by a matrix P such that the resulting 
matrix is orthogonal, i.e. 
(PT)(PT)' s I 
Hence if the matrix equation to be solved is: 
T™ = k 
one has x = {PT)'y 
where y 7» Pk, which solves the problem, 
But, as (PT) is formed row by row, it follows that (PT)! is 
formed column by column while Pk is formed element by element. 
Hence if one accumulates: En 
e th.element of Pk)x (i th, column of (PT) *)| 
] 
as one proceeds, one will end with having the required column x 
without the necessity of a back solution, 
2. If i and j are non-orthogonal n-vectors, so that i.i x90, 
consider the vector: 1 
| 1.4 
Piu once t | 
* Bi] 
| T J 
iis is clearly now orthogonal to i. By combining "ows o? T in 
l ^ 
is way, which corresponds to premultiplying by a row of P, or 
rather t hefirst of a series of elementary matrices which will 
end up as P, one can clearly make every row of PT orthogonal to 
every other, so that the matrix is orthogonal, 
3. The above is standard procedure; the new proposal is thats 
when one has a quasi-band matrix such as the above, where the 
blank parts of the matrix consist of all zeros, one should proceed 
as follows:- 
  
  
  
  
cui