WikiMath

In [16]: def trigonalise(A, Y):

    ...:     n = A.shape[0]
    ...:     for i in range(n-1):
    ...:         for k in range(i+1,n):
    ...:             c = A[k,i]/A[i,i]
    ...:             transvection(A,Y,k,i,c)
    ...: 

In [17]: A = matrix([ [1, 2, 2], [1, 3, -2], [3, 5, 8] ])

In [18]: Y=matrix([ [2], [-1], [8] ])

In [19]: trigonalise(A,Y)

In [20]: A Out[20]:

matrix([ [ 1, 2, 2],

        [ 0,  1, -4],
        [ 0,  0, -2] ])