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| from .utilities import _iszero | |
| def _columnspace(M, simplify=False): | |
| """Returns a list of vectors (Matrix objects) that span columnspace of ``M`` | |
| Examples | |
| ======== | |
| >>> from sympy import Matrix | |
| >>> M = Matrix(3, 3, [1, 3, 0, -2, -6, 0, 3, 9, 6]) | |
| >>> M | |
| Matrix([ | |
| [ 1, 3, 0], | |
| [-2, -6, 0], | |
| [ 3, 9, 6]]) | |
| >>> M.columnspace() | |
| [Matrix([ | |
| [ 1], | |
| [-2], | |
| [ 3]]), Matrix([ | |
| [0], | |
| [0], | |
| [6]])] | |
| See Also | |
| ======== | |
| nullspace | |
| rowspace | |
| """ | |
| reduced, pivots = M.echelon_form(simplify=simplify, with_pivots=True) | |
| return [M.col(i) for i in pivots] | |
| def _nullspace(M, simplify=False, iszerofunc=_iszero): | |
| """Returns list of vectors (Matrix objects) that span nullspace of ``M`` | |
| Examples | |
| ======== | |
| >>> from sympy import Matrix | |
| >>> M = Matrix(3, 3, [1, 3, 0, -2, -6, 0, 3, 9, 6]) | |
| >>> M | |
| Matrix([ | |
| [ 1, 3, 0], | |
| [-2, -6, 0], | |
| [ 3, 9, 6]]) | |
| >>> M.nullspace() | |
| [Matrix([ | |
| [-3], | |
| [ 1], | |
| [ 0]])] | |
| See Also | |
| ======== | |
| columnspace | |
| rowspace | |
| """ | |
| reduced, pivots = M.rref(iszerofunc=iszerofunc, simplify=simplify) | |
| free_vars = [i for i in range(M.cols) if i not in pivots] | |
| basis = [] | |
| for free_var in free_vars: | |
| # for each free variable, we will set it to 1 and all others | |
| # to 0. Then, we will use back substitution to solve the system | |
| vec = [M.zero] * M.cols | |
| vec[free_var] = M.one | |
| for piv_row, piv_col in enumerate(pivots): | |
| vec[piv_col] -= reduced[piv_row, free_var] | |
| basis.append(vec) | |
| return [M._new(M.cols, 1, b) for b in basis] | |
| def _rowspace(M, simplify=False): | |
| """Returns a list of vectors that span the row space of ``M``. | |
| Examples | |
| ======== | |
| >>> from sympy import Matrix | |
| >>> M = Matrix(3, 3, [1, 3, 0, -2, -6, 0, 3, 9, 6]) | |
| >>> M | |
| Matrix([ | |
| [ 1, 3, 0], | |
| [-2, -6, 0], | |
| [ 3, 9, 6]]) | |
| >>> M.rowspace() | |
| [Matrix([[1, 3, 0]]), Matrix([[0, 0, 6]])] | |
| """ | |
| reduced, pivots = M.echelon_form(simplify=simplify, with_pivots=True) | |
| return [reduced.row(i) for i in range(len(pivots))] | |
| def _orthogonalize(cls, *vecs, normalize=False, rankcheck=False): | |
| """Apply the Gram-Schmidt orthogonalization procedure | |
| to vectors supplied in ``vecs``. | |
| Parameters | |
| ========== | |
| vecs | |
| vectors to be made orthogonal | |
| normalize : bool | |
| If ``True``, return an orthonormal basis. | |
| rankcheck : bool | |
| If ``True``, the computation does not stop when encountering | |
| linearly dependent vectors. | |
| If ``False``, it will raise ``ValueError`` when any zero | |
| or linearly dependent vectors are found. | |
| Returns | |
| ======= | |
| list | |
| List of orthogonal (or orthonormal) basis vectors. | |
| Examples | |
| ======== | |
| >>> from sympy import I, Matrix | |
| >>> v = [Matrix([1, I]), Matrix([1, -I])] | |
| >>> Matrix.orthogonalize(*v) | |
| [Matrix([ | |
| [1], | |
| [I]]), Matrix([ | |
| [ 1], | |
| [-I]])] | |
| See Also | |
| ======== | |
| MatrixBase.QRdecomposition | |
| References | |
| ========== | |
| .. [1] https://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process | |
| """ | |
| from .decompositions import _QRdecomposition_optional | |
| if not vecs: | |
| return [] | |
| all_row_vecs = (vecs[0].rows == 1) | |
| vecs = [x.vec() for x in vecs] | |
| M = cls.hstack(*vecs) | |
| Q, R = _QRdecomposition_optional(M, normalize=normalize) | |
| if rankcheck and Q.cols < len(vecs): | |
| raise ValueError("GramSchmidt: vector set not linearly independent") | |
| ret = [] | |
| for i in range(Q.cols): | |
| if all_row_vecs: | |
| col = cls(Q[:, i].T) | |
| else: | |
| col = cls(Q[:, i]) | |
| ret.append(col) | |
| return ret | |