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