| 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: |
| |
| |
| 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 |
|
|
