Because I really needed sparse operations, I recoded the right_kernel
function as follows (I need the output basis). I realize after reading
your code that these two are actually equivalent - you use functions
dealing with pivots while I directly fetch them from the matrix (I did
not know the existence of these functions).
def right_kernel_basis(m):
field=m.base_ring()
lead_positions=[]
row_vector_space=VectorSpace(m.base_ring(), m.ncols(),
sparse=True)
m_ech=m.echelon_form()
additional_columns=[]
last_j = -1
last_row=-1
for i in xrange(m_ech.nrows()):
nz_pos = m_ech.row(i).nonzero_positions()
if len(nz_pos)==0:
last_row=i
break
j = nz_pos[0]
lead_positions.append(j)
assert j>last_j # echelon form
assert m_ech[i][j] == 1
if last_j >= 0:
additional_columns += range(last_j+1, j)
last_j = j
if last_row<0:
last_row = m_ech.nrows()
ker_vectors=[]
for j in additional_columns + range(last_j+1, m_ech.ncols()):
col = m_ech.column(j)
v = row_vector_space(0)
v[j] = 1
for i in col.nonzero_positions():
v[lead_positions[i]] = -col[i]
assert m*v == 0
ker_vectors.append(v)
assert len(ker_vectors) + last_row == m.ncols()
if len(ker_vectors)==0:
return []
mret = Matrix(field, ker_vectors, sparse=True).transpose()
assert mret.rank() == len(ker_vectors)
assert m*mret == 0
return ker_vectors
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