ai-content-maker/.venv/Lib/site-packages/sympy/matrices/tests/test_subspaces.py

115 lines
3.7 KiB
Python

from sympy.matrices.common import _MinimalMatrix, _CastableMatrix
from sympy.matrices.matrices import MatrixSubspaces
from sympy.matrices import Matrix
from sympy.core.numbers import Rational
from sympy.core.symbol import symbols
from sympy.solvers import solve
class SubspaceOnlyMatrix(_MinimalMatrix, _CastableMatrix, MatrixSubspaces):
pass
# SubspaceOnlyMatrix tests
def test_columnspace_one():
m = SubspaceOnlyMatrix([[ 1, 2, 0, 2, 5],
[-2, -5, 1, -1, -8],
[ 0, -3, 3, 4, 1],
[ 3, 6, 0, -7, 2]])
basis = m.columnspace()
assert basis[0] == Matrix([1, -2, 0, 3])
assert basis[1] == Matrix([2, -5, -3, 6])
assert basis[2] == Matrix([2, -1, 4, -7])
assert len(basis) == 3
assert Matrix.hstack(m, *basis).columnspace() == basis
def test_rowspace():
m = SubspaceOnlyMatrix([[ 1, 2, 0, 2, 5],
[-2, -5, 1, -1, -8],
[ 0, -3, 3, 4, 1],
[ 3, 6, 0, -7, 2]])
basis = m.rowspace()
assert basis[0] == Matrix([[1, 2, 0, 2, 5]])
assert basis[1] == Matrix([[0, -1, 1, 3, 2]])
assert basis[2] == Matrix([[0, 0, 0, 5, 5]])
assert len(basis) == 3
def test_nullspace_one():
m = SubspaceOnlyMatrix([[ 1, 2, 0, 2, 5],
[-2, -5, 1, -1, -8],
[ 0, -3, 3, 4, 1],
[ 3, 6, 0, -7, 2]])
basis = m.nullspace()
assert basis[0] == Matrix([-2, 1, 1, 0, 0])
assert basis[1] == Matrix([-1, -1, 0, -1, 1])
# make sure the null space is really gets zeroed
assert all(e.is_zero for e in m*basis[0])
assert all(e.is_zero for e in m*basis[1])
def test_nullspace_second():
# first test reduced row-ech form
R = Rational
M = Matrix([[5, 7, 2, 1],
[1, 6, 2, -1]])
out, tmp = M.rref()
assert out == Matrix([[1, 0, -R(2)/23, R(13)/23],
[0, 1, R(8)/23, R(-6)/23]])
M = Matrix([[-5, -1, 4, -3, -1],
[ 1, -1, -1, 1, 0],
[-1, 0, 0, 0, 0],
[ 4, 1, -4, 3, 1],
[-2, 0, 2, -2, -1]])
assert M*M.nullspace()[0] == Matrix(5, 1, [0]*5)
M = Matrix([[ 1, 3, 0, 2, 6, 3, 1],
[-2, -6, 0, -2, -8, 3, 1],
[ 3, 9, 0, 0, 6, 6, 2],
[-1, -3, 0, 1, 0, 9, 3]])
out, tmp = M.rref()
assert out == Matrix([[1, 3, 0, 0, 2, 0, 0],
[0, 0, 0, 1, 2, 0, 0],
[0, 0, 0, 0, 0, 1, R(1)/3],
[0, 0, 0, 0, 0, 0, 0]])
# now check the vectors
basis = M.nullspace()
assert basis[0] == Matrix([-3, 1, 0, 0, 0, 0, 0])
assert basis[1] == Matrix([0, 0, 1, 0, 0, 0, 0])
assert basis[2] == Matrix([-2, 0, 0, -2, 1, 0, 0])
assert basis[3] == Matrix([0, 0, 0, 0, 0, R(-1)/3, 1])
# issue 4797; just see that we can do it when rows > cols
M = Matrix([[1, 2], [2, 4], [3, 6]])
assert M.nullspace()
def test_columnspace_second():
M = Matrix([[ 1, 2, 0, 2, 5],
[-2, -5, 1, -1, -8],
[ 0, -3, 3, 4, 1],
[ 3, 6, 0, -7, 2]])
# now check the vectors
basis = M.columnspace()
assert basis[0] == Matrix([1, -2, 0, 3])
assert basis[1] == Matrix([2, -5, -3, 6])
assert basis[2] == Matrix([2, -1, 4, -7])
#check by columnspace definition
a, b, c, d, e = symbols('a b c d e')
X = Matrix([a, b, c, d, e])
for i in range(len(basis)):
eq=M*X-basis[i]
assert len(solve(eq, X)) != 0
#check if rank-nullity theorem holds
assert M.rank() == len(basis)
assert len(M.nullspace()) + len(M.columnspace()) == M.cols