def test_inverse_non_invertible():
raises(NonSquareMatrixError, lambda: Inverse(A))
raises(NonSquareMatrixError, lambda: Inverse(A*B))
raises(NonSquareMatrixError, lambda: ZeroMatrix(n, m).I)
raises(NonInvertibleMatrixError, lambda: ZeroMatrix(n, n).I)
raises(NonSquareMatrixError, lambda: OneMatrix(n, m).I)
raises(NonInvertibleMatrixError, lambda: OneMatrix(2, 2).I)
def test_BlockMatrix_2x2_inverse_symbolic():
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', n, k - m)
C = MatrixSymbol('C', k - n, m)
D = MatrixSymbol('D', k - n, k - m)
X = BlockMatrix([[A, B], [C, D]])
assert X.is_square and X.shape == (k, k)
assert isinstance(block_collapse(
X.I), Inverse) # Can't invert when none of the blocks is square
# test code path where only A is invertible
A = MatrixSymbol('A', n, n)
B = MatrixSymbol('B', n, m)
C = MatrixSymbol('C', m, n)
D = ZeroMatrix(m, m)
X = BlockMatrix([[A, B], [C, D]])
assert block_collapse(X.inverse()) == BlockMatrix([
[A.I + A.I * B * X.schur('A').I * C * A.I, -A.I * B * X.schur('A').I],
[-X.schur('A').I * C * A.I, X.schur('A').I],
])
# test code path where only B is invertible
A = MatrixSymbol('A', n, m)
B = MatrixSymbol('B', n, n)
C = ZeroMatrix(m, m)
D = MatrixSymbol('D', m, n)
X = BlockMatrix([[A, B], [C, D]])
assert block_collapse(X.inverse()) == BlockMatrix([
[-X.schur('B').I * D * B.I, X.schur('B').I],
[B.I + B.I * A * X.schur('B').I * D * B.I, -B.I * A * X.schur('B').I],
])
# test code path where only C is invertible
A = MatrixSymbol('A', n, m)
B = ZeroMatrix(n, n)
C = MatrixSymbol('C', m, m)
D = MatrixSymbol('D', m, n)
X = BlockMatrix([[A, B], [C, D]])
assert block_collapse(X.inverse()) == BlockMatrix([
[-C.I * D * X.schur('C').I, C.I + C.I * D * X.schur('C').I * A * C.I],
[X.schur('C').I, -X.schur('C').I * A * C.I],
])
# test code path where only D is invertible
A = ZeroMatrix(n, n)
B = MatrixSymbol('B', n, m)
C = MatrixSymbol('C', m, n)
D = MatrixSymbol('D', m, m)
X = BlockMatrix([[A, B], [C, D]])
assert block_collapse(X.inverse()) == BlockMatrix([
[X.schur('D').I, -X.schur('D').I * B * D.I],
[-D.I * C * X.schur('D').I, D.I + D.I * C * X.schur('D').I * B * D.I],
])
def test_triangular():
assert ask(Q.upper_triangular(X + Z.T + Identity(2)),
Q.upper_triangular(X) & Q.lower_triangular(Z)) is True
assert ask(Q.upper_triangular(X * Z.T),
Q.upper_triangular(X) & Q.lower_triangular(Z)) is True
assert ask(Q.lower_triangular(Identity(3))) is True
assert ask(Q.lower_triangular(ZeroMatrix(3, 3))) is True
assert ask(Q.upper_triangular(ZeroMatrix(3, 3))) is True
assert ask(Q.lower_triangular(OneMatrix(1, 1))) is True
assert ask(Q.upper_triangular(OneMatrix(1, 1))) is True
assert ask(Q.lower_triangular(OneMatrix(3, 3))) is False
assert ask(Q.upper_triangular(OneMatrix(3, 3))) is False
assert ask(Q.triangular(X), Q.unit_triangular(X))
assert ask(Q.upper_triangular(X**3), Q.upper_triangular(X))
assert ask(Q.lower_triangular(X**3), Q.lower_triangular(X))
def test_trace():
assert isinstance(Trace(A), Trace)
assert not isinstance(Trace(A), MatrixExpr)
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)
A / Trace(A) # Make sure this is possible
# 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).doit() == (0 + 0) + (1 + 1) + (2 + 2)
raises(TypeError, lambda: Trace(S.One))
assert Trace(A).arg is A
def test_eval_determinant():
assert det(Identity(n)) == 1
assert det(ZeroMatrix(n, n)) == 0
assert det(OneMatrix(n, n)) == Determinant(OneMatrix(n, n))
assert det(OneMatrix(1, 1)) == 1
assert det(OneMatrix(2, 2)) == 0
assert det(Transpose(A)) == det(A)
def test_invertible_BlockMatrix():
assert ask(Q.invertible(BlockMatrix([Identity(3)]))) == True
assert ask(Q.invertible(BlockMatrix([ZeroMatrix(3, 3)]))) == False
X = Matrix([[1, 2, 3], [3, 5, 4]])
Y = Matrix([[4, 2, 7], [2, 3, 5]])
# non-invertible A block
assert ask(
Q.invertible(
BlockMatrix([
[Matrix.ones(3, 3), Y.T],
[X, Matrix.eye(2)],
]))) == True
# non-invertible B block
assert ask(
Q.invertible(
BlockMatrix([
[Y.T, Matrix.ones(3, 3)],
[Matrix.eye(2), X],
]))) == True
# non-invertible C block
assert ask(
Q.invertible(
BlockMatrix([
[X, Matrix.eye(2)],
[Matrix.ones(3, 3), Y.T],
]))) == True
# non-invertible D block
assert ask(
Q.invertible(
BlockMatrix([
[Matrix.eye(2), X],
[Y.T, Matrix.ones(3, 3)],
]))) == True
def test_diagonal():
assert ask(Q.diagonal(X + Z.T + Identity(2)), Q.diagonal(X) &
Q.diagonal(Z)) is True
assert ask(Q.diagonal(ZeroMatrix(3, 3)))
assert ask(Q.lower_triangular(X) & Q.upper_triangular(X), Q.diagonal(X))
assert ask(Q.diagonal(X), Q.lower_triangular(X) & Q.upper_triangular(X))
assert ask(Q.symmetric(X), Q.diagonal(X))
assert ask(Q.triangular(X), Q.diagonal(X))
def test_sympy__matrices__expressions__blockmatrix__BlockMatrix():
from sympy.matrices.expressions.blockmatrix import BlockMatrix
from sympy.matrices.expressions import MatrixSymbol, ZeroMatrix
X = MatrixSymbol('X', x, x)
Y = MatrixSymbol('Y', y, y)
Z = MatrixSymbol('Z', x, y)
O = ZeroMatrix(y, x)
assert _test_args(BlockMatrix([[X, Z], [O, Y]]))
def test_matrix_element_sets():
X = MatrixSymbol('X', 4, 4)
assert ask(Q.real(X[1, 2]), Q.real_elements(X))
assert ask(Q.integer(X[1, 2]), Q.integer_elements(X))
assert ask(Q.complex(X[1, 2]), Q.complex_elements(X))
assert ask(Q.integer_elements(Identity(3)))
assert ask(Q.integer_elements(ZeroMatrix(3, 3)))
from sympy.matrices.expressions.fourier import DFT
assert ask(Q.complex_elements(DFT(3)))
def test_fullrank():
assert ask(Q.fullrank(X), Q.fullrank(X))
assert ask(Q.fullrank(X.T), Q.fullrank(X)) is True
assert ask(Q.fullrank(X)) is None
assert ask(Q.fullrank(Y)) is None
assert ask(Q.fullrank(X*Z), Q.fullrank(X) & Q.fullrank(Z)) is True
assert ask(Q.fullrank(Identity(3))) is True
assert ask(Q.fullrank(ZeroMatrix(3, 3))) is False
assert ask(Q.invertible(X), ~Q.fullrank(X)) == False
def test_BlockMatrix_Determinant():
A, B, C, D = [MatrixSymbol(s, 3, 3) for s in 'ABCD']
X = BlockMatrix([[A, B], [C, D]])
from sympy import assuming, Q
with assuming(Q.invertible(A)):
assert det(X) == det(A) * det(D - C * A.I * B)
assert isinstance(det(X), Expr)
assert det(BlockMatrix([A])) == det(A)
assert det(BlockMatrix([ZeroMatrix(n, n)])) == 0
def _test_orthogonal_unitary(predicate):
assert ask(predicate(X), predicate(X))
assert ask(predicate(X.T), predicate(X)) is True
assert ask(predicate(X.I), predicate(X)) is True
assert ask(predicate(Y)) is False
assert ask(predicate(X)) is None
assert ask(predicate(X*Z*X), predicate(X) & predicate(Z)) is True
assert ask(predicate(Identity(3))) is True
assert ask(predicate(ZeroMatrix(3, 3))) is False
assert ask(Q.invertible(X), predicate(X))
assert not ask(predicate(X + Z), predicate(X) & predicate(Z))
def test_BlockMatrix_Determinant():
A, B, C, D = [MatrixSymbol(s, 3, 3) for s in 'ABCD']
X = BlockMatrix([[A, B], [C, D]])
from sympy.assumptions.ask import Q
from sympy.assumptions.assume import assuming
with assuming(Q.invertible(A)):
assert det(X) == det(A) * det(X.schur('A'))
assert isinstance(det(X), Expr)
assert det(BlockMatrix([A])) == det(A)
assert det(BlockMatrix([ZeroMatrix(n, n)])) == 0
def test_invertible():
assert ask(Q.invertible(X), Q.invertible(X))
assert ask(Q.invertible(Y)) is False
assert ask(Q.invertible(X*Y), Q.invertible(X)) is False
assert ask(Q.invertible(X*Z), Q.invertible(X)) is None
assert ask(Q.invertible(X*Z), Q.invertible(X) & Q.invertible(Z)) is True
assert ask(Q.invertible(X.T)) is None
assert ask(Q.invertible(X.T), Q.invertible(X)) is True
assert ask(Q.invertible(X.I)) is True
assert ask(Q.invertible(Identity(3))) is True
assert ask(Q.invertible(ZeroMatrix(3, 3))) is False
assert ask(Q.invertible(X), Q.fullrank(X) & Q.square(X))
def test_BlockDiagMatrix_trace():
assert trace(BlockDiagMatrix()) == 0
assert trace(BlockDiagMatrix(ZeroMatrix(n, n))) == 0
A = MatrixSymbol('A', n, n)
assert trace(BlockDiagMatrix(A)) == trace(A)
B = MatrixSymbol('B', m, m)
assert trace(BlockDiagMatrix(A, B)) == trace(A) + trace(B)
# non-square blocks
C = MatrixSymbol('C', m, n)
D = MatrixSymbol('D', n, m)
assert isinstance(trace(BlockDiagMatrix(C, D)), Trace)
def test_invalid_block_matrix():
raises(ValueError, lambda: BlockMatrix([
[Identity(2), Identity(5)],
]))
raises(ValueError, lambda: BlockMatrix([
[Identity(n), Identity(m)],
]))
raises(ValueError, lambda: BlockMatrix([
[ZeroMatrix(n, n), ZeroMatrix(n, n)],
[ZeroMatrix(n, n - 1), ZeroMatrix(n, n + 1)],
]))
raises(ValueError, lambda: BlockMatrix([
[ZeroMatrix(n - 1, n), ZeroMatrix(n, n)],
[ZeroMatrix(n + 1, n), ZeroMatrix(n, n)],
]))
def test_issue_21866():
n = 10
I = Identity(n)
O = ZeroMatrix(n, n)
A = BlockMatrix([[ I, O, O, O ],
[ O, I, O, O ],
[ O, O, I, O ],
[ I, O, O, I ]])
Ainv = block_collapse(A.inv())
AinvT = BlockMatrix([[ I, O, O, O ],
[ O, I, O, O ],
[ O, O, I, O ],
[ -I, O, O, I ]])
assert Ainv == AinvT
def test_diagonal():
assert ask(Q.diagonal(X + Z.T + Identity(2)),
Q.diagonal(X) & Q.diagonal(Z)) is True
assert ask(Q.diagonal(ZeroMatrix(3, 3)))
assert ask(Q.lower_triangular(X) & Q.upper_triangular(X), Q.diagonal(X))
assert ask(Q.diagonal(X), Q.lower_triangular(X) & Q.upper_triangular(X))
assert ask(Q.symmetric(X), Q.diagonal(X))
assert ask(Q.triangular(X), Q.diagonal(X))
assert ask(Q.diagonal(C0x0))
assert ask(Q.diagonal(A1x1))
assert ask(Q.diagonal(A1x1 + B1x1))
assert ask(Q.diagonal(A1x1 * B1x1))
assert ask(Q.diagonal(V1.T * V2))
assert ask(Q.diagonal(V1.T * (X + Z) * V1))
assert ask(Q.diagonal(MatrixSlice(Y, (0, 1), (1, 2)))) is True
assert ask(Q.diagonal(V1.T * (V1 + V2))) is True
def test_positive_definite():
assert ask(Q.positive_definite(X), Q.positive_definite(X))
assert ask(Q.positive_definite(X.T), Q.positive_definite(X)) is True
assert ask(Q.positive_definite(X.I), Q.positive_definite(X)) is True
assert ask(Q.positive_definite(Y)) is False
assert ask(Q.positive_definite(X)) is None
assert ask(Q.positive_definite(X*Z*X),
Q.positive_definite(X) & Q.positive_definite(Z)) is True
assert ask(Q.positive_definite(X), Q.orthogonal(X))
assert ask(Q.positive_definite(Y.T*X*Y),
Q.positive_definite(X) & Q.fullrank(Y)) is True
assert not ask(Q.positive_definite(Y.T*X*Y), Q.positive_definite(X))
assert ask(Q.positive_definite(Identity(3))) is True
assert ask(Q.positive_definite(ZeroMatrix(3, 3))) is False
assert ask(Q.positive_definite(X + Z), Q.positive_definite(X) &
Q.positive_definite(Z)) is True
assert not ask(Q.positive_definite(-X), Q.positive_definite(X))
assert ask(Q.positive(X[1, 1]), Q.positive_definite(X))
def test_symmetric():
assert ask(Q.symmetric(X), Q.symmetric(X))
assert ask(Q.symmetric(X * Z), Q.symmetric(X)) is None
assert ask(Q.symmetric(X * Z), Q.symmetric(X) & Q.symmetric(Z)) is True
assert ask(Q.symmetric(X + Z), Q.symmetric(X) & Q.symmetric(Z)) is True
assert ask(Q.symmetric(Y)) is False
assert ask(Q.symmetric(Y * Y.T)) is True
assert ask(Q.symmetric(Y.T * X * Y)) is None
assert ask(Q.symmetric(Y.T * X * Y), Q.symmetric(X)) is True
assert ask(Q.symmetric(X**10), Q.symmetric(X)) is True
assert ask(Q.symmetric(A1x1)) is True
assert ask(Q.symmetric(A1x1 + B1x1)) is True
assert ask(Q.symmetric(A1x1 * B1x1)) is True
assert ask(Q.symmetric(V1.T * V1)) is True
assert ask(Q.symmetric(V1.T * (V1 + V2))) is True
assert ask(Q.symmetric(V1.T * (V1 + V2) + A1x1)) is True
assert ask(Q.symmetric(MatrixSlice(Y, (0, 1), (1, 2)))) is True
assert ask(Q.symmetric(Identity(3))) is True
assert ask(Q.symmetric(ZeroMatrix(3, 3))) is True
assert ask(Q.symmetric(OneMatrix(3, 3))) is True
def test_BlockMatrix_trace():
A, B, C, D = [MatrixSymbol(s, 3, 3) for s in 'ABCD']
X = BlockMatrix([[A, B], [C, D]])
assert trace(X) == trace(A) + trace(D)
assert trace(BlockMatrix([ZeroMatrix(n, n)])) == 0
def test_sympy__matrices__expressions__matexpr__ZeroMatrix():
from sympy.matrices.expressions.matexpr import ZeroMatrix
assert _test_args(ZeroMatrix(3, 5))
def test_invertible_BlockDiagMatrix():
assert ask(Q.invertible(BlockDiagMatrix(Identity(3), Identity(5)))) == True
assert ask(Q.invertible(BlockDiagMatrix(ZeroMatrix(3, 3),
Identity(5)))) == False
assert ask(Q.invertible(BlockDiagMatrix(Identity(3),
OneMatrix(5, 5)))) == False
def test_issue_17624():
a = MatrixSymbol("a", 2, 2)
z = ZeroMatrix(2, 2)
b = BlockMatrix([[a, z], [z, z]])
assert block_collapse(b * b) == BlockMatrix([[a**2, z], [z, z]])
assert block_collapse(b * b * b) == BlockMatrix([[a**3, z], [z, z]])