Vehicle-cpp/hkpc/software/sympy/assumptions/tests/test_matrices.py

from sympy.assumptions.ask import (Q, ask)
from sympy.core.symbol import Symbol
from sympy.matrices.expressions.diagonal import (DiagMatrix, DiagonalMatrix)
from sympy.matrices.dense import Matrix
from sympy.matrices.expressions import (MatrixSymbol, Identity, ZeroMatrix,
        OneMatrix, Trace, MatrixSlice, Determinant, BlockMatrix, BlockDiagMatrix)
from sympy.matrices.expressions.factorizations import LofLU
from sympy.testing.pytest import XFAIL

X = MatrixSymbol('X', 2, 2)
Y = MatrixSymbol('Y', 2, 3)
Z = MatrixSymbol('Z', 2, 2)
A1x1 = MatrixSymbol('A1x1', 1, 1)
B1x1 = MatrixSymbol('B1x1', 1, 1)
C0x0 = MatrixSymbol('C0x0', 0, 0)
V1 = MatrixSymbol('V1', 2, 1)
V2 = MatrixSymbol('V2', 2, 1)

def test_square():
    assert ask(Q.square(X))
    assert not ask(Q.square(Y))
    assert ask(Q.square(Y*Y.T))

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(OneMatrix(1, 1))) is True
    assert ask(Q.invertible(OneMatrix(3, 3))) is False
    assert ask(Q.invertible(X), Q.fullrank(X) & Q.square(X))

def test_singular():
    assert ask(Q.singular(X)) is None
    assert ask(Q.singular(X), Q.invertible(X)) is False
    assert ask(Q.singular(X), ~Q.invertible(X)) is True

@XFAIL
def test_invertible_fullrank():
    assert ask(Q.invertible(X), Q.fullrank(X)) is True


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_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_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_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(X**2), predicate(X))
    assert ask(predicate(Y)) is False
    assert ask(predicate(X)) is None
    assert ask(predicate(X), ~Q.invertible(X)) is False
    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_orthogonal():
    _test_orthogonal_unitary(Q.orthogonal)

def test_unitary():
    _test_orthogonal_unitary(Q.unitary)
    assert ask(Q.unitary(X), Q.orthogonal(X))

def test_fullrank():
    assert ask(Q.fullrank(X), Q.fullrank(X))
    assert ask(Q.fullrank(X**2), 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.fullrank(OneMatrix(1, 1))) is True
    assert ask(Q.fullrank(OneMatrix(3, 3))) is False
    assert ask(Q.invertible(X), ~Q.fullrank(X)) == False


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**3), Q.positive_definite(X))
    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(OneMatrix(1, 1))) is True
    assert ask(Q.positive_definite(OneMatrix(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_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_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.diagonal(OneMatrix(1, 1))) is True
    assert ask(Q.diagonal(OneMatrix(3, 3))) is False
    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
    assert ask(Q.diagonal(X**3), Q.diagonal(X))
    assert ask(Q.diagonal(Identity(3)))
    assert ask(Q.diagonal(DiagMatrix(V1)))
    assert ask(Q.diagonal(DiagonalMatrix(X)))


def test_non_atoms():
    assert ask(Q.real(Trace(X)), Q.positive(Trace(X)))

@XFAIL
def test_non_trivial_implies():
    X = MatrixSymbol('X', 3, 3)
    Y = MatrixSymbol('Y', 3, 3)
    assert ask(Q.lower_triangular(X+Y), Q.lower_triangular(X) &
               Q.lower_triangular(Y)) is True
    assert ask(Q.triangular(X), Q.lower_triangular(X)) is True
    assert ask(Q.triangular(X+Y), Q.lower_triangular(X) &
               Q.lower_triangular(Y)) is True

def test_MatrixSlice():
    X = MatrixSymbol('X', 4, 4)
    B = MatrixSlice(X, (1, 3), (1, 3))
    C = MatrixSlice(X, (0, 3), (1, 3))
    assert ask(Q.symmetric(B), Q.symmetric(X))
    assert ask(Q.invertible(B), Q.invertible(X))
    assert ask(Q.diagonal(B), Q.diagonal(X))
    assert ask(Q.orthogonal(B), Q.orthogonal(X))
    assert ask(Q.upper_triangular(B), Q.upper_triangular(X))

    assert not ask(Q.symmetric(C), Q.symmetric(X))
    assert not ask(Q.invertible(C), Q.invertible(X))
    assert not ask(Q.diagonal(C), Q.diagonal(X))
    assert not ask(Q.orthogonal(C), Q.orthogonal(X))
    assert not ask(Q.upper_triangular(C), Q.upper_triangular(X))

def test_det_trace_positive():
    X = MatrixSymbol('X', 4, 4)
    assert ask(Q.positive(Trace(X)), Q.positive_definite(X))
    assert ask(Q.positive(Determinant(X)), Q.positive_definite(X))

def test_field_assumptions():
    X = MatrixSymbol('X', 4, 4)
    Y = MatrixSymbol('Y', 4, 4)
    assert ask(Q.real_elements(X), Q.real_elements(X))
    assert not ask(Q.integer_elements(X), Q.real_elements(X))
    assert ask(Q.complex_elements(X), Q.real_elements(X))
    assert ask(Q.complex_elements(X**2), Q.real_elements(X))
    assert ask(Q.real_elements(X**2), Q.integer_elements(X))
    assert ask(Q.real_elements(X+Y), Q.real_elements(X)) is None
    assert ask(Q.real_elements(X+Y), Q.real_elements(X) & Q.real_elements(Y))
    from sympy.matrices.expressions.hadamard import HadamardProduct
    assert ask(Q.real_elements(HadamardProduct(X, Y)),
                    Q.real_elements(X) & Q.real_elements(Y))
    assert ask(Q.complex_elements(X+Y), Q.real_elements(X) & Q.complex_elements(Y))

    assert ask(Q.real_elements(X.T), Q.real_elements(X))
    assert ask(Q.real_elements(X.I), Q.real_elements(X) & Q.invertible(X))
    assert ask(Q.real_elements(Trace(X)), Q.real_elements(X))
    assert ask(Q.integer_elements(Determinant(X)), Q.integer_elements(X))
    assert not ask(Q.integer_elements(X.I), Q.integer_elements(X))
    alpha = Symbol('alpha')
    assert ask(Q.real_elements(alpha*X), Q.real_elements(X) & Q.real(alpha))
    assert ask(Q.real_elements(LofLU(X)), Q.real_elements(X))
    e = Symbol('e', integer=True, negative=True)
    assert ask(Q.real_elements(X**e), Q.real_elements(X) & Q.invertible(X))
    assert ask(Q.real_elements(X**e), Q.real_elements(X)) is None

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)))
    assert ask(Q.integer_elements(OneMatrix(3, 3)))
    from sympy.matrices.expressions.fourier import DFT
    assert ask(Q.complex_elements(DFT(3)))


def test_matrix_element_sets_slices_blocks():
    X = MatrixSymbol('X', 4, 4)
    assert ask(Q.integer_elements(X[:, 3]), Q.integer_elements(X))
    assert ask(Q.integer_elements(BlockMatrix([[X], [X]])),
                        Q.integer_elements(X))

def test_matrix_element_sets_determinant_trace():
    assert ask(Q.integer(Determinant(X)), Q.integer_elements(X))
    assert ask(Q.integer(Trace(X)), Q.integer_elements(X))