U Zʹj@sBddlmZd ddZdefddZdddZddd d d Zd S))_iszeroFcs$j|dd\}}fdd|DS)aReturns 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 TsimplifyZ with_pivotscsg|]}|qS)col.0iMrW/var/www/html/bwave/python/venv/lib/python3.8/site-packages/sympy/matrices/subspaces.py #sz _columnspace..) echelon_form)r rreducedpivotsrr r _columnspacesrc sj||d\}fddtjD}g}|D]P}jgj}j||<tD] \}} || |||f8<qV||q2fdd|DS)aReturns 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 ) iszerofuncrcsg|]}|kr|qSrrr)rrr r Bsz_nullspace..csg|]}jd|qS)r)_newcols)rbr rr r Ps)Zrrefrangerzeroone enumerateappend) r rrrZ free_varsZbasisZfree_varvecZpiv_rowZpiv_colr)r rr _nullspace&s  rcs,|j|dd\}fddtt|DS)aDReturns 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]])] Trcsg|]}|qSr)rowrrrr r fsz_rowspace..)rrlen)r rrrrr _rowspaceSsr ) normalize rankcheckc Gsddlm}|sgS|djdk}dd|D}|j|}|||d\}}|rd|jt|krdtdg} t|jD]>} |r||dd| fj} n||dd| f} | | qr| S) aApply 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 r)_QRdecomposition_optionalcSsg|] }|qSr)r)rxrrr r sz"_orthogonalize..)r!z0GramSchmidt: vector set not linearly independentN) Zdecompositionsr#rowsZhstackrr ValueErrorrTr) clsr!r"Zvecsr#Z all_row_vecsr QRretr rrrr _orthogonalizeis 0   r-N)F)F)Z utilitiesrrrr r-rrrr s  "-