Untitled

Here is a matrix with both lower and upper bandwidth equal to one. Such a matrix is called tridiagonal.

A = [ 2 -1  0  0  0  0
2 -1  0  0  0
3  0 -1  0  0
0  2  2 -1  0
0  0  1  1 -1
0  0  0  0  2 ]
A = 
  -1     0     0     0     0
   2    -1     0     0     0
   3     0    -1     0     0
   0     2     2    -1     0
   0     0     1     1    -1
   0     0     0     0     2

We can extract the elements on any diagonal using the diag command. The "main'' or central diagonal is numbered zero, above and to the right of that is positive, and below and to the left is negative.

diag_main = diag(A,0)'
diag_main = 
     2     2     0     2     1     2

diag_plusone = diag(A,1)'
diag_plusone = 
    -1    -1    -1    -1    -1

diag_minusone = diag(A,-1)'
diag_minusone = 
     4     3     2     1     0

We can also put whatever numbers we like onto any diagonal with the diag command.

A = A + diag([5 8 6 7],2)
A = 
  -1     5     0     0     0
   2    -1     8     0     0
   3     0    -1     6     0
   0     2     2    -1     7
   0     0     1     1    -1
   0     0     0     0     2

Here is what happens when we factor this matrix without pivoting.

[L,U] = lufact(A);

subplot(1,2,1), spy(L), title('L')

subplot(1,2,2), spy(U), title('U')

Observe that the factors preserve the lower and upper bandwidth of the original matrix. However, if we introduce row pivoting, this structure may be destroyed.

[L,U,P] = lu(A);

subplot(1,2,1), spy(L), title('L')

subplot(1,2,2), spy(U), title('U')