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