def test_echelon_form():
# echelon form is not unique, but the result
# must be row-equivalent to the original matrix
# and it must be in echelon form.
a = zeros_Reductions(3)
e = eye_Reductions(3)
# we can assume the zero matrix and the identity matrix shouldn't change
assert a.echelon_form() == a
assert e.echelon_form() == e
a = ReductionsOnlyMatrix(0, 0, [])
assert a.echelon_form() == a
a = ReductionsOnlyMatrix(1, 1, [5])
assert a.echelon_form() == a
# now we get to the real tests
def verify_row_null_space(mat, rows, nulls):
for v in nulls:
assert all(t.is_zero for t in a_echelon * v)
for v in rows:
if not all(t.is_zero for t in v):
assert not all(t.is_zero for t in a_echelon * v.transpose())
a = ReductionsOnlyMatrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])
nulls = [Matrix([[1], [-2], [1]])]
rows = [a[i, :] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
a = ReductionsOnlyMatrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 8])
nulls = []
rows = [a[i, :] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
a = ReductionsOnlyMatrix(3, 3, [2, 1, 3, 0, 0, 0, 2, 1, 3])
nulls = [Matrix([[-1 / 2], [1], [0]]), Matrix([[-3 / 2], [0], [1]])]
rows = [a[i, :] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
# this one requires a row swap
a = ReductionsOnlyMatrix(3, 3, [2, 1, 3, 0, 0, 0, 1, 1, 3])
nulls = [Matrix([[0], [-3], [1]])]
rows = [a[i, :] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
a = ReductionsOnlyMatrix(3, 3, [0, 3, 3, 0, 2, 2, 0, 1, 1])
nulls = [Matrix([[1], [0], [0]]), Matrix([[0], [-1], [1]])]
rows = [a[i, :] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
a = ReductionsOnlyMatrix(2, 3, [2, 2, 3, 3, 3, 0])
nulls = [Matrix([[-1], [1], [0]])]
rows = [a[i, :] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
def test_echelon_form():
# echelon form is not unique, but the result
# must be row-equivalent to the original matrix
# and it must be in echelon form.
a = zeros_Reductions(3)
e = eye_Reductions(3)
# we can assume the zero matrix and the identity matrix shouldn't change
assert a.echelon_form() == a
assert e.echelon_form() == e
a = ReductionsOnlyMatrix(0, 0, [])
assert a.echelon_form() == a
a = ReductionsOnlyMatrix(1, 1, [5])
assert a.echelon_form() == a
# now we get to the real tests
def verify_row_null_space(mat, rows, nulls):
for v in nulls:
assert all(t.is_zero for t in a_echelon*v)
for v in rows:
if not all(t.is_zero for t in v):
assert not all(t.is_zero for t in a_echelon*v.transpose())
a = ReductionsOnlyMatrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])
nulls = [Matrix([
[ 1],
[-2],
[ 1]])]
rows = [a[i,:] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
a = ReductionsOnlyMatrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 8])
nulls = []
rows = [a[i,:] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
a = ReductionsOnlyMatrix(3, 3, [2, 1, 3, 0, 0, 0, 2, 1, 3])
nulls = [Matrix([
[-S(1)/2],
[ 1],
[ 0]]),
Matrix([
[-S(3)/2],
[ 0],
[ 1]])]
rows = [a[i,:] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
# this one requires a row swap
a = ReductionsOnlyMatrix(3, 3, [2, 1, 3, 0, 0, 0, 1, 1, 3])
nulls = [Matrix([
[ 0],
[ -3],
[ 1]])]
rows = [a[i,:] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
a = ReductionsOnlyMatrix(3, 3, [0, 3, 3, 0, 2, 2, 0, 1, 1])
nulls = [Matrix([
[1],
[0],
[0]]),
Matrix([
[ 0],
[-1],
[ 1]])]
rows = [a[i,:] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)
a = ReductionsOnlyMatrix(2, 3, [2, 2, 3, 3, 3, 0])
nulls = [Matrix([
[-1],
[1],
[0]])]
rows = [a[i,:] for i in range(a.rows)]
a_echelon = a.echelon_form()
assert a_echelon.is_echelon
verify_row_null_space(a, rows, nulls)