1909.sagews

n=3;
variables=[]
for row in range(n):
    variables.append([]);
    for column in range(n):
        variables[row].append(var('z'+str(n-row)+str(column+1)))
variables

[[z31, z32, z33], [z21, z22, z23], [z11, z12, z13]]

p=[2,1,3]
for column in range (n):
    variables[p[column]-1][column]=1
    for column_prime in range (column+1,n):
        variables[p[column]-1][column_prime]=0

M = matrix(variables);M

[z31 1 0] [ 1 0 0] [z11 z12 1]

e=[]
zero_vector=list(0 for dummy in range(n))
for i in range (n):
    zero_vector[i]=1;
    e.append(vector(zero_vector));
    zero_vector[i]=0;
e

[(1, 0, 0), (0, 1, 0), (0, 0, 1)]

s=e[1:]
s.append(vector(zero_vector))
N=matrix(s)
N

[0 1 0] [0 0 1] [0 0 0]

a=[];
for row in range(n):
    a.append([])
    for column in range(row+2):
        a[row].append(var('a'+str(row+1)+str(column+1)))
a

[[a11, a12], [a21, a22, a23], [a31, a32, a33, a34]]

X=[];
for i in range(n-2):
    X.append([]);
    X[i]=vector(zero_vector)
    for j in range (i+2):
        X[i]=X[i]+a[i][j]*M*e[j]
X

[(a11*z31 + a12, a11, a11*z11 + a12*z12)]

for j in range(n-2):
    for i in range(n):
        print(e[i]*N*M*e[j],e[i]*X[j])

(1, a11*z31 + a12) (z11, a11) (0, a11*z11 + a12*z12)

for j in range(n-2):
    for i in range(n):
        print(e[i]*X[j],e[i]*N*M*e[j])

(a11*z31 + a12, 1) (a11, z11) (a11*z11 + a12*z12, 0)