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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
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