Please look at the first cell in this worksheet.
The Problem
Wanted to simplify the following symbolic calculation: If S be a matrix, nXn, of symbols - the matrix $A_{ab}=\sum_{c\ne{d}}S_{ac}S_{ad}S_{bd}S_{bc}$.
Construction of the matrix of symbols
n = 3
R = PolynomialRing(RR,[['r','t'][cmp(int(i/n),i%n)]+'_'+str(1+int(i/n))+str(1+i%n) for i in range(n^2)])
S = matrix(R,3,R.gens())
And this works, the matrix is constructed.
sanity Check
print [S[int(k/n)][k%n] for k in range(n^2)]
works perfectly.
Roadblock
However this construct does not work:
print [(S[i][int(k/n)]S[i][k%n]S[j][k%n]S[j][int(k/n)])(int(k/n)!=k%n)for k in range(n^2)]
and returns
Traceback (most recent call last):
File "", line 1, in <module>
File "/tmp/tmpQltfcX/___code___.py", line 10, in <module>
print [(S[i][int(k/n)]*S[i][k%n]*S[j][k%n]*S[j][int(k/n)])*(int(k/n)!=k%n)for k in range(n**_sage_const_2 )]
File "matrix0.pyx", line 924, in sage.matrix.matrix0.Matrix.__getitem__ (sage/matrix/matrix0.c:5540)
IndexError: matrix index out of range
Consequently the constructor for the matrix $A_{ab}$ doesnt work as well!
Check Again
print flatten([[[(i,int(k/n),k%n,j,1*(int(k/n)!=k%n))for k in range(n^2)]for i in range(n)]for j in range(n)])
returns a list containing 0, 1 and 2 s only. So the IndexError up there does not make sense either!
Edit: Addendum
n = 3
R = PolynomialRing(RR,[['r','t'][cmp(int(i/n),i%n)]+'_'+str(1+int(i/n))+str(1+i%n) for i in range(n^2)])
S = matrix(R,3,R.gens())
from operator import add
def Sh(i,j):
return reduce(add,[S[i][k//n]*S[i][k%n]*S[j][k%n]*S[j][k//n]*((k//n)!=k%n) for k in range(n^2)])
Shot = matrix(RR,n,n, Sh)
#print Shot
sage then returns the following error:
Traceback (most recent call last): #print flatten([[[(i,int(k/n),k%n,j,1*(int(k/n)!=k%n))for k in range(n^2)]for i in range(n)]for j in range(n)])
File "", line 1, in <module>
File "/tmp/tmpGXJyv5/___code___.py", line 12, in <module>
Shot = matrix(RR,n,n, Sh)
File "/sagenb/sage_install/sage-4.8-sage.math.washington.edu-x86_64-Linux/local/lib/python2.6/site-packages/sage/matrix/constructor.py", line 666, in matrix
return matrix_space.MatrixSpace(ring, nrows, ncols, sparse=sparse)(entries)
File "/sagenb/sage_install/sage-4.8-sage.math.washington.edu-x86_64-Linux/local/lib/python2.6/site-packages/sage/matrix/matrix_space.py", line 462, in __call__
return self.matrix(entries, copy=copy, coerce=coerce, rows=rows)
File "/sagenb/sage_install/sage-4.8-sage.math.washington.edu-x86_64-Linux/local/lib/python2.6/site-packages/sage/matrix/matrix_space.py", line 1266, in matrix
return self.__matrix_class(self, entries=x, copy=copy, coerce=coerce)
File "matrix_generic_dense.pyx", line 109, in sage.matrix.matrix_generic_dense.Matrix_generic_dense.__init__ (sage/matrix/matrix_generic_dense.c:2411)
File "parent.pyx", line 988, in sage.structure.parent.Parent.__call__ (sage/structure/parent.c:7355)
File "polynomial_element.pyx", line 6316, in sage.rings.polynomial.polynomial_element.ConstantPolynomialSection._call_ (sage/rings/polynomial/polynomial_element.c:43471)
TypeError: not a constant polynomial
