def test_as_explicit_nonsquare():
A = ImmutableMatrix([[1, 2, 3], [4, 5, 6]])
assert MatPow(A, 1).as_explicit() == A
pytest.raises(ShapeError, lambda: MatPow(A, 0).as_explicit())
pytest.raises(ShapeError, lambda: MatPow(A, 2).as_explicit())
pytest.raises(ShapeError, lambda: MatPow(A, -1).as_explicit())
pytest.raises(ValueError, lambda: MatPow(A, pi).as_explicit())
def test_entry():
assert MatPow(A, 1)[0, 0] == A[0, 0]
assert MatPow(C, 0)[0, 0] == 1
assert MatPow(C, 0)[0, 1] == 0
assert isinstance(MatPow(C, 2)[0, 0], Sum)
pytest.raises(NotImplementedError, lambda: MatPow(C, -1)[0, 0])
def test_doit_nonsquare():
X = ImmutableMatrix([[1, 2, 3], [4, 5, 6]])
assert MatPow(X, 1).doit() == X
pytest.raises(ShapeError, lambda: MatPow(X, 0).doit())
pytest.raises(ShapeError, lambda: MatPow(X, 2).doit())
pytest.raises(ShapeError, lambda: MatPow(X, -1).doit())
pytest.raises(ShapeError, lambda: MatPow(X, pi).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_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_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_doit_args():
A = ImmutableMatrix([[1, 2], [3, 4]])
B = ImmutableMatrix([[2, 3], [4, 5]])
assert MatAdd(A, MatPow(B, 2)).doit() == A + B**2
assert MatAdd(A, MatMul(A, B)).doit() == A + A * B
assert MatAdd(A, A).doit(deep=False) == 2 * A
assert (MatAdd(A, X, MatMul(A, B), Y,
MatAdd(2 * A, B)).doit() == MatAdd(X, Y, 3 * A + A * B + B))
def test_Trace_doit_deep_False():
X = Matrix([[1, 2], [3, 4]])
assert Trace(X).doit(deep=False) == 5
q = MatPow(X, 2)
assert Trace(q).doit(deep=False).arg == q
q = MatAdd(X, 2 * X)
assert Trace(q).doit(deep=False).arg == q
q = MatMul(X, 2 * X)
assert Trace(q).doit(deep=False).arg == q
def test_identity_power():
k = Identity(n)
assert MatPow(k, 4).doit() == k
assert MatPow(k, n).doit() == k
assert MatPow(k, -3).doit() == k
assert MatPow(k, 0).doit() == k
l = Identity(3)
assert MatPow(l, n).doit() == l
assert MatPow(l, -1).doit() == l
assert MatPow(l, 0).doit() == l
def test_identity_powers():
M = Identity(n)
assert MatPow(M, 3).doit() == M**3
assert M**n == M
assert MatPow(M, 0).doit() == M**2
assert M**-2 == M
assert MatPow(M, -2).doit() == M**0
N = Identity(3)
assert MatPow(N, 2).doit() == N**n
assert MatPow(N, 3).doit() == N
assert MatPow(N, -2).doit() == N**4
assert MatPow(N, 2).doit() == N**0
def test_MatPow():
A = MatrixSymbol('A', n, n)
AA = MatPow(A, 2)
assert AA.exp == 2
assert AA.base == A
assert (A**n).exp == n
assert A**0 == Identity(n)
assert A**1 == A
assert A**2 == AA
assert A**-1 == Inverse(A)
assert A**Rational(1, 2) == sqrt(A)
pytest.raises(ShapeError, lambda: MatrixSymbol('B', 3, 2)**2)
def test_as_explicit():
A = ImmutableMatrix([[1, 2], [3, 4]])
assert MatPow(A, 0).as_explicit() == ImmutableMatrix(Identity(2))
assert MatPow(A, 1).as_explicit() == A
assert MatPow(A, 2).as_explicit() == A**2
assert MatPow(A, -1).as_explicit() == A.inv()
assert MatPow(A, -2).as_explicit() == (A.inv())**2
# less expensive than testing on a 2x2
A = ImmutableMatrix([4])
assert MatPow(A, Rational(1, 2)).as_explicit() == A**Rational(1, 2)
def test_doit_with_MatrixBase():
X = ImmutableMatrix([[1, 2], [3, 4]])
assert MatPow(X, 0).doit() == ImmutableMatrix(Identity(2))
assert MatPow(X, 1).doit() == X
assert MatPow(X, 2).doit() == X**2
assert MatPow(X, -1).doit() == X.inv()
assert MatPow(X, -2).doit() == (X.inv())**2
# less expensive than testing on a 2x2
assert MatPow(ImmutableMatrix([4]),
Rational(1, 2)).doit() == ImmutableMatrix([2])
def test_pow_index():
Q = MatPow(A, 2)
assert Q[0, 0] == A[0, 0]**2 + A[0, 1]*A[1, 0]
def test_matpow():
pytest.raises(TypeError, lambda: MatPow(1, 1))
def test_as_explicit_symbol():
X = MatrixSymbol('X', 2, 2)
assert MatPow(X, 0).as_explicit() == ImmutableMatrix(Identity(2))
assert MatPow(X, 1).as_explicit() == X.as_explicit()
assert MatPow(X, 2).as_explicit() == (X.as_explicit())**2
def test_as_explicit_nonsquare_symbol():
X = MatrixSymbol('X', 2, 3)
assert MatPow(X, 1).as_explicit() == X.as_explicit()
for r in [0, 2, Rational(1, 2), pi]:
pytest.raises(ShapeError, lambda: MatPow(X, r).as_explicit())
def test_Trace_MatPow_doit():
X = Matrix([[1, 2], [3, 4]])
assert Trace(X).doit() == 5
q = MatPow(X, 2)
assert Trace(q).arg == q
assert Trace(q).doit() == 29
def test_doit_drills_down():
X = ImmutableMatrix([[1, 2], [3, 4]])
Y = ImmutableMatrix([[2, 3], [4, 5]])
assert MatMul(X, MatPow(Y, 2)).doit() == X * Y**2
assert MatMul(C, Transpose(D * C)).doit().args == (C, C.T, D.T)
def test_doit_nested_MatrixExpr():
X = ImmutableMatrix([[1, 2], [3, 4]])
Y = ImmutableMatrix([[2, 3], [4, 5]])
assert MatPow(MatMul(X, Y), 2).doit() == (X * Y)**2
assert MatPow(MatAdd(X, Y), 2).doit() == (X + Y)**2
def test_doit_nonsquare_MatrixSymbol():
assert MatPow(A, 1).doit() == A
for r in [0, 2, -1, pi]:
assert MatPow(A, r).doit() == MatPow(A, r)
def test_doit_square_MatrixSymbol_symsize():
assert MatPow(C, 0).doit() == Identity(n)
assert MatPow(C, 0).doit(deep=False) == Identity(n)
assert MatPow(C, 1).doit() == C
for r in [2, -1, pi]:
assert MatPow(C, r).doit() == MatPow(C, r)