def test_matexpr():
assert (x*A).shape == A.shape
assert (x*A).__class__ == MatMul
assert 2*A - A - A == ZeroMatrix(*A.shape)
assert (A*B).shape == (n, l)
assert A.equals(ZeroMatrix(3, 3)) is None
# issue diofant/diofant#469
expr = Eq(C, D)
assert simplify(expr) == expr
def test_zero_power():
z1 = ZeroMatrix(n, n)
assert MatPow(z1, 3).doit() == z1
pytest.raises(ValueError, lambda: MatPow(z1, -1).doit())
assert MatPow(z1, 0).doit() == Identity(n)
assert MatPow(z1, n).doit() == z1
pytest.raises(ValueError, lambda: MatPow(z1, -2).doit())
z2 = ZeroMatrix(4, 4)
assert MatPow(z2, n).doit() == z2
pytest.raises(ValueError, lambda: MatPow(z2, -3).doit())
assert MatPow(z2, 2).doit() == z2
assert MatPow(z2, 0).doit() == Identity(4)
pytest.raises(ValueError, lambda: MatPow(z2, -1).doit())
def test_Zero_power():
z1 = ZeroMatrix(n, n)
assert z1**4 == z1
pytest.raises(ValueError, lambda: z1**-2)
assert z1**0 == Identity(n)
assert MatPow(z1, 2).doit() == z1**2
pytest.raises(ValueError, lambda: MatPow(z1, -2).doit())
z2 = ZeroMatrix(3, 3)
assert MatPow(z2, 4).doit() == z2**4
pytest.raises(ValueError, lambda: z2**-3)
assert z2**3 == MatPow(z2, 3).doit()
assert z2**0 == Identity(3)
pytest.raises(ValueError, lambda: MatPow(z2, -1).doit())
def test_BlockMatrix():
n, m = symbols('n m', integer=True)
X = MatrixSymbol('X', n, n)
Y = MatrixSymbol('Y', m, m)
Z = MatrixSymbol('Z', n, m)
B = BlockMatrix([[X, Z], [ZeroMatrix(m, n), Y]])
assert str(B) == "Matrix([\n[X, Z],\n[0, Y]])"
def test_addition():
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', n, m)
assert isinstance(A + B, MatAdd)
assert (A + B).shape == A.shape
assert isinstance(A - A + 2*B, MatMul)
pytest.raises(ShapeError, lambda: A + B.T)
pytest.raises(TypeError, lambda: A + 1)
pytest.raises(TypeError, lambda: 5 + A)
pytest.raises(TypeError, lambda: 5 - A)
assert A + ZeroMatrix(n, m) - A == ZeroMatrix(n, m)
with pytest.raises(TypeError):
ZeroMatrix(n, m) + Integer(0) # pylint: disable=expression-not-assigned
def test_Trace():
assert isinstance(Trace(A), Trace)
assert not isinstance(Trace(A), MatrixExpr)
pytest.raises(ShapeError, lambda: Trace(C))
assert trace(eye(3)) == 3
assert trace(Matrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])) == 15
assert adjoint(Trace(A)) == trace(Adjoint(A))
assert conjugate(Trace(A)) == trace(Adjoint(A))
assert transpose(Trace(A)) == Trace(A)
assert isinstance(A / Trace(A), MatrixExpr)
# Some easy simplifications
assert trace(Identity(5)) == 5
assert trace(ZeroMatrix(5, 5)) == 0
assert trace(2 * A * B) == 2 * Trace(A * B)
assert trace(A.T) == trace(A)
i, j = symbols('i j')
F = FunctionMatrix(3, 3, Lambda((i, j), i + j))
assert trace(F) == (0 + 0) + (1 + 1) + (2 + 2)
pytest.raises(TypeError, lambda: Trace(1))
assert Trace(A).arg is A
assert str(trace(A)) == str(Trace(A).doit())
def test_invariants():
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', m, l)
X = MatrixSymbol('X', n, n)
objs = [Identity(n), ZeroMatrix(m, n), MatMul(A, B), MatAdd(A, A),
Transpose(A), Adjoint(A), Inverse(X), MatPow(X, 2), MatPow(X, -1),
MatPow(X, 0)]
for obj in objs:
assert obj == obj.__class__(*obj.args)
def test_multiplication():
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', m, l)
C = MatrixSymbol('C', n, n)
assert (2*A*B).shape == (n, l)
assert (A*0*B) == ZeroMatrix(n, l)
pytest.raises(ShapeError, lambda: B*A)
assert (2*A).shape == A.shape
assert A * ZeroMatrix(m, m) * B == ZeroMatrix(n, l)
assert C * Identity(n) * C.inverse() == Identity(n)
assert B/2 == Rational(1, 2)*B
pytest.raises(NotImplementedError, lambda: 2/B)
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', n, n)
assert Identity(n) * (A + B) == A + B
def test_block_index():
I = Identity(3)
Z = ZeroMatrix(3, 3)
B = BlockMatrix([[I, I], [I, I]])
e3 = ImmutableMatrix(eye(3))
BB = BlockMatrix([[e3, e3], [e3, e3]])
assert B[0, 0] == B[3, 0] == B[0, 3] == B[3, 3] == 1
assert B[4, 3] == B[5, 1] == 0
BB = BlockMatrix([[e3, e3], [e3, e3]])
assert B.as_explicit() == BB.as_explicit()
BI = BlockMatrix([[I, Z], [Z, I]])
assert BI.as_explicit().equals(eye(6))
def test_ZeroMatrix():
A = MatrixSymbol('A', n, m)
Z = ZeroMatrix(n, m)
assert A + Z == A
assert A*Z.T == ZeroMatrix(n, n)
assert Z*A.T == ZeroMatrix(n, n)
assert A - A == ZeroMatrix(*A.shape)
assert not Z
assert transpose(Z) == ZeroMatrix(m, n)
assert Z.conjugate() == Z
assert ZeroMatrix(n, n)**0 == Identity(n)
with pytest.raises(ShapeError):
Z**0 # pylint: disable=pointless-statement
with pytest.raises(ShapeError):
Z**2 # pylint: disable=pointless-statement
def test_eval_determinant():
assert det(Matrix()) == 1
assert det(Identity(n)) == 1
assert det(ZeroMatrix(n, n)) == 0
assert det(Transpose(A)) == det(A)
def test_ZeroMatrix_doit():
Znn = ZeroMatrix(Add(n, n, evaluate=False), n)
assert isinstance(Znn.rows, Add)
assert Znn.doit() == ZeroMatrix(2*n, n)
assert isinstance(Znn.doit().rows, Mul)
def test_any_zeros():
assert any_zeros(MatMul(A, ZeroMatrix(m, k), evaluate=False)) == \
ZeroMatrix(n, k)
def test_matexpr():
assert (x * A).shape == A.shape
assert (x * A).__class__ == MatMul
assert 2 * A - A - A == ZeroMatrix(*A.shape)
assert (A * B).shape == (n, l)