1920.sagews

n=5;
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

[[z51, z52, z53, z54, z55], [z41, z42, z43, z44, z45], [z31, z32, z33, z34, z35], [z21, z22, z23, z24, z25], [z11, z12, z13, z14, z15]]

p=[3,2,4,1,5]
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
    for row in range (p[column],n):
        variables[row][column]=0

M = matrix(variables);M

[z51 z52 z53 1 0] [z41 1 0 0 0] [ 1 0 0 0 0] [ 0 0 1 0 0] [ 0 0 0 0 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, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)]

s=[]
for row in range (n):
    s.append(e[row]*(row+1));
    
N=matrix(s)
N

[1 0 0 0 0] [0 2 0 0 0] [0 0 3 0 0] [0 0 0 4 0] [0 0 0 0 5]

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], [a41, a42, a43, a44, a45], [a51, a52, a53, a54, a55, a56]]

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*z51 + a12*z52, a11*z41 + a12, a11, 0, 0), (a21*z51 + a22*z52 + a23*z53, a21*z41 + a22, a21, a23, 0), (a31*z51 + a32*z52 + a33*z53 + a34, a31*z41 + a32, a31, a33, 0)]

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

(z51, a11*z51 + a12*z52) (2*z41, a11*z41 + a12) (3, a11) (0, 0) (0, 0) (z52, a21*z51 + a22*z52 + a23*z53) (2, a21*z41 + a22) (0, a21) (0, a23) (0, 0) (z53, a31*z51 + a32*z52 + a33*z53 + a34) (0, a31*z41 + a32) (0, a31) (4, a33) (0, 0)

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

(a11*z51 + a12*z52, z51) (a11*z41 + a12, 2*z41) (a11, 3) (0, 0) (0, 0) (a21*z51 + a22*z52 + a23*z53, z52) (a21*z41 + a22, 2) (a21, 0) (a23, 0) (0, 0) (a31*z51 + a32*z52 + a33*z53 + a34, z53) (a31*z41 + a32, 0) (a31, 0) (a33, 4) (0, 0)