+ i[^RIHt^RIHt^RIHt^RIHt^RIH t H t H t ^RI H t HtHtHtHtHtHtHtHtHtHtHtHtHtHt^RIHtHtHtH t ^RI!H"t"H#t#H$t$H%t%^R I&H't'H(t(H)t)H*t*H+t+H,t,H-t-H.t.H/t/H0t0H1t1H2t2H3t3H4t4^R I5H6t6RR ./t7!R R ]4t8!RR]84t9!RR]94t:!RR]:4t;!RR]4t<!RR]4t=R#))Basic)Dummy) MatrixCommon)NonSquareMatrixError)_iszero_is_zero_after_expand_mul _simplify)_find_reasonable_pivot_find_reasonable_pivot_naive _adjugate _charpoly _cofactor_cofactor_matrix_per_det _det_bareiss_det_berkowitz _det_bird _det_laplace_det_LU_minor_minor_submatrix) _is_echelon _echelon_form_rank_rref) _columnspace _nullspace _rowspace_orthogonalize) _eigenvals _eigenvects_bidiagonalize_bidiagonal_decomposition_is_diagonalizable _diagonalize_is_positive_definite_is_positive_semidefinite_is_negative_definite_is_negative_semidefinite_is_indefinite _jordan_form_left_eigenvects_singular_values) MatrixBase matplotlibca]tRt^/toRt]3RltRt]R3Rlt Rt Rt Rt RR lt R ]3R ltRR ltRR ltRRltRtRRltRt]P]n]P]n]P]n]P]n]P] n]P] n]P] n]P] n]P] n]P]n] P]n]!P]n]P]n]"P]n]#P]n]$P]nRt%Vt&R#)MatrixDeterminantzProvides basic matrix determinant operations. Should not be instantiated directly. See ``determinant.py`` for their implementations.c\WR7#)) iszerofuncr)selfr4s&&w/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/matrices/matrices.py_eval_det_bareiss#MatrixDeterminant._eval_det_bareiss3s D88c\V4#N)rr6s&r7_eval_det_berkowitz%MatrixDeterminant._eval_det_berkowitz6s d##r:Nc\WVR7#))r4simpfunc)r)r6r4rAs&&&r7 _eval_det_luMatrixDeterminant._eval_det_lu9stXFFr:c\V4#r<)rr=s&r7_eval_det_bird MatrixDeterminant._eval_det_bird<s r:c\V4#r<)rr=s&r7_eval_det_laplace#MatrixDeterminant._eval_det_laplace? D!!r:c\V4#r<rr=s&r7_eval_determinant#MatrixDeterminant._eval_determinantB Dzr:c\WR7#method)r r6rSs&&r7adjugateMatrixDeterminant.adjugateEs --r:lambdac\WVR7#))xsimplify)r r6rYrZs&&&r7charpolyMatrixDeterminant.charpolyHsX66r:c\WW#R7#rQ)rr6ijrSs&&&&r7cofactorMatrixDeterminant.cofactorKs!33r:c\WR7#rQ)rrTs&&r7cofactor_matrix!MatrixDeterminant.cofactor_matrixNs 44r:c\WVR7#))rSr4rL)r6rSr4s&&&r7detMatrixDeterminant.detQsDJ??r:c\V4#r<)rr=s&r7perMatrixDeterminant.perTrOr:c\WW#R7#rQ)rr_s&&&&r7minorMatrixDeterminant.minorWsdq00r:c\WV4#r<)rr6r`ras&&&r7minor_submatrix!MatrixDeterminant.minor_submatrixZs++r: berkowitz)bareissN)'__name__ __module__ __qualname____firstlineno____doc__rr8r>rrBrErHrMrUr r\rbrerhrkrnrrr r rrrrrrr r rrrrr__static_attributes____classdictcell__ __classdict__s@r7r2r2/sFC,E9$'.G"."I745@1,,B+I+I"+G+O+O (+7+?+?+9+A+A(1(9(9N+7+?+?+2??L+/<<+4+<+>EM+;+C+COr:r2c4a]tRt^otoRt]RR3Rlt]R4t]R3Rlt Rt ]RRR3Rlt ] P]n] P]n]P] n]P] nRR ltR tR tR tR tRtRtRRltRRltRtVtR#)MatrixReductionszProvides basic matrix row/column operations. Should not be instantiated directly. See ``reductions.py`` for some of their implementations.Fc\WVVR7#))r4rZ with_pivots)r)r6r4rZrs&&&&r7 echelon_formMatrixReductions.echelon_formssT8') )r:c\V4#r<)rr=s&r7 is_echelonMatrixReductions.is_echelonws 4  r:c\WVR7#))r4rZ)r)r6r4rZs&&&r7rankMatrixReductions.rank{sT8DDr:c \VPWPVP4V44wr#VRRVP13,VRVP)R13,3#)a]Return reduced row-echelon form of matrix, matrix showing rhs after reduction steps. ``rhs`` must have the same number of rows as ``self``. Examples ======== >>> from sympy import Matrix, symbols >>> r1, r2 = symbols('r1 r2') >>> Matrix([[1, 1], [2, 1]]).rref_rhs(Matrix([r1, r2])) (Matrix([ [1, 0], [0, 1]]), Matrix([ [ -r1 + r2], [2*r1 - r2]])) :NNNN)rhstackeyerowscols)r6rhsr_s&& r7rref_rhsMatrixReductions.rref_rhs~sU"T[[xx ':C@AJTYYJ1sxxij=!111r:Tc\WVW4R7#))r4rZpivotsnormalize_last)r)r6r4rZrrs&&&&&r7rrefMatrixReductions.rrefsT8: :r:c nVR 9d\RPWa44hVR8Xd VPM VPpVR8XdXVeTMTpVeVf\RPV44h^Tu;8:dV8gM\RPWb44hEMVR8XdW#WE0P R.4p\ V4^8dW$V0P R.4p\ V4^8wd\R PV44hVwrE^Tu;8:dV8gM\RPWd44h^Tu;8:dV8gM\RPWe44hMVR8XdVfTMTpVfTMTpVe VeVf\R PV44hW%8Xd\R PV44h^Tu;8:dV8gM\RPWb44h^Tu;8:dV8gM\RPWe44hM\R \ V4,4hWW4V3#)zValidate the arguments for a row/column operation. ``error_str`` can be one of "row" or "col" depending on the arguments being parsed.n->knn<->mn->n+kmzOUnknown {} operation '{}'. Valid col operations are 'n->kn', 'n<->m', 'n->n+km'colNzEFor a {0} operation 'n->kn' you must provide the kwargs `{0}` and `k`z#This matrix does not have a {} '{}'zIFor a {0} operation 'n<->m' you must provide the kwargs `{0}1` and `{0}2`zPFor a {0} operation 'n->n+km' you must provide the kwargs `{0}`, `k`, and `{0}2`zAFor a {0} operation 'n->n+km' `{0}` and `{0}2` must be different.zinvalid operation %s)rrr) ValueErrorformatrr differencelenrepr) r6oprkcol1col2 error_str self_colsrs &&&&&&& r7_normalize_op_args#MatrixReductions._normalize_op_argssG 2 2??Evi?TV V"+e!3DII  =#dC{ai "88>y8IKK'i' !F!M!Mi!]^^(7]D'22D6:D4y1}4(33TF;4yA~ "<? ?%%r:cfaaaVVV3RlpSPSPSPV4#)cH<VS8XdSSW3,,#SW3,#r<rt)r`rarrr6s&&r7entryBMatrixReductions._eval_col_op_multiply_col_by_const..entry&Cx4:~%: r:_newrr)r6rrrsfff r7"_eval_col_op_multiply_col_by_const3MatrixReductions._eval_col_op_multiply_col_by_const% yyDIIu55r:cfaaaVVV3RlpSPSPSPV4#)c`<VS8Xd SVS3,#VS8Xd SVS3,#SW3,#r<rt)r`rarrr6s&&r7r1MatrixReductions._eval_col_op_swap..entrys9DyAtG}$dAtG}$: r:r)r6rrrsfff r7_eval_col_op_swap"MatrixReductions._eval_col_op_swap%  yyDIIu55r:cjaaaaVVVV3RlpSPSPSPV4#)ch<VS8Xd"SW3,SSVS3,,,#SW3,#r<rt)r`rarrrr6s&&r7rFMatrixReductions._eval_col_op_add_multiple_to_other_col..entrys4CxADzAQW $555: r:r)r6rrrrsffff r7&_eval_col_op_add_multiple_to_other_col7MatrixReductions._eval_col_op_add_multiple_to_other_col*  yyDIIu55r:cfaaaVVV3RlpSPSPSPV4#)c`<VS8Xd SSV3,#VS8Xd SSV3,#SW3,#r<rt)r`rarow1row2r6s&&r7r1MatrixReductions._eval_row_op_swap..entrys9DyD!G}$dD!G}$: r:r)r6rrrsfff r7_eval_row_op_swap"MatrixReductions._eval_row_op_swaprr:cfaaaVVV3RlpSPSPSPV4#)cH<VS8XdSSW3,,#SW3,#r<rt)r`rarrowr6s&&r7rBMatrixReductions._eval_row_op_multiply_row_by_const..entryrr:r)r6rrrsfff r7"_eval_row_op_multiply_row_by_const3MatrixReductions._eval_row_op_multiply_row_by_constrr:cjaaaaVVVV3RlpSPSPSPV4#)ch<VS8Xd"SW3,SSSV3,,,#SW3,#r<rt)r`rarrrr6s&&r7rFMatrixReductions._eval_row_op_add_multiple_to_other_row..entrys4CxADzAT1W $555: r:r)r6rrrrsffff r7&_eval_row_op_add_multiple_to_other_row7MatrixReductions._eval_row_op_add_multiple_to_other_rowrr:Nc VPWW4VR4wrr4pVR8XdVPW#4#VR8XdVPWE4#VR8XdVPW#V4#R#)aPerforms the elementary column operation `op`. `op` may be one of * ``"n->kn"`` (column n goes to k*n) * ``"n<->m"`` (swap column n and column m) * ``"n->n+km"`` (column n goes to column n + k*column m) Parameters ========== op : string; the elementary row operation col : the column to apply the column operation k : the multiple to apply in the column operation col1 : one column of a column swap col2 : second column of a column swap or column "m" in the column operation "n->n+km" rrrrN)rrrr)r6rrrrrs&&&&&&r7elementary_col_op"MatrixReductions.elementary_col_ops("&!8!8!4QV!W$ =::3B B =))$5 5 ?>>stL L r:c VPWW4VR4wrr4pVR8XdVPW#4#VR8XdVPWE4#VR8XdVPW#V4#R#)aPerforms the elementary row operation `op`. `op` may be one of * ``"n->kn"`` (row n goes to k*n) * ``"n<->m"`` (swap row n and row m) * ``"n->n+km"`` (row n goes to row n + k*row m) Parameters ========== op : string; the elementary row operation row : the row to apply the row operation k : the multiple to apply in the row operation row1 : one row of a row swap row2 : second row of a row swap or row "m" in the row operation "n->n+km" rrrrN)rrrr)r6rrrrrs&&&&&&r7elementary_row_op"MatrixReductions.elementary_row_oprr:rt)r)rNNNN)rxryrzr{r|rrpropertyrrrrrrrrrrrrrrrrrr}r~rs@r7rrosJ'.5)!!&E2(&d: )00L&..J ==DL ==DL5&n666666M<MMr:rca]tRtRtoRtR RltR]3RltR RltRt ] P]n] P]n] P]n] P] n]!] 4t RtVtR #) MatrixSubspacesi>zProvides methods relating to the fundamental subspaces of a matrix. Should not be instantiated directly. See ``subspaces.py`` for their implementations.Fc\WR7#)rZ)rr6rZs&&r7 columnspaceMatrixSubspaces.columnspaceCs D44r:c\WVR7#))rZr4)r)r6rZr4s&&&r7 nullspaceMatrixSubspaces.nullspaceFs$jIIr:c\WR7#r)rrs&&r7rowspaceMatrixSubspaces.rowspaceIs 11r:c \V.VO5/VB#r<)r )clsvecskwargss&*,r7 orthogonalizeMatrixSubspaces.orthogonalizeOsc3D3F33r:rtNF)rxryrzr{r|rrrrrrrrr classmethodr}r~rs@r7rr>sj5"'7J2 4)00K&..I%--H*22M' 6Mr:rca]tRtRtoRtRRltR]3RltRRltRRlt RRlt RR lt ] R 4t ] R 4t] R 4t] R 4t] R4tRRltRtRt]P]n]P]n]P]n]P] n]P] n]P]n]P]n]P]n]P]n]P]n]P]n] P]n]!P] n]"P] nRt#Vt$R#) MatrixEigeniZzProvides basic matrix eigenvalue/vector operations. Should not be instantiated directly. See ``eigen.py`` for their implementations.Tc \V3RV/VB#)error_when_incomplete)r!)r6rflagss&&,r7 eigenvalsMatrixEigen.eigenvals_s$U6KUuUUr:c "\V3RVRV/VB#)rr4)r")r6rr4rs&&&,r7 eigenvectsMatrixEigen.eigenvectsbs)407L0%0).0 0r:c \V3RV/VB#) reals_only)r%)r6rrs&&,r7is_diagonalizableMatrixEigen.is_diagonalizablefs!$H:HHHr:c\WVVR7#))rsort normalize)r&)r6rrrs&&&&r7 diagonalizeMatrixEigen.diagonalizeisDd#% %r:c\WR7#)upper)r#r6rs&&r7 bidiagonalizeMatrixEigen.bidiagonalizems d00r:c\WR7#r)r$r s&&r7bidiagonal_decomposition$MatrixEigen.bidiagonal_decompositionps (;;r:c\V4#r<)r'r=s&r7is_positive_definite MatrixEigen.is_positive_definites $T**r:c\V4#r<)r(r=s&r7is_positive_semidefinite$MatrixEigen.is_positive_semidefinitew (..r:c\V4#r<)r)r=s&r7is_negative_definite MatrixEigen.is_negative_definite{rr:c\V4#r<)r*r=s&r7is_negative_semidefinite$MatrixEigen.is_negative_semidefiniterr:c\V4#r<)r+r=s&r7 is_indefiniteMatrixEigen.is_indefinites d##r:c \V3RV/VB#)calc_transform)r,)r6r!rs&&,r7 jordan_formMatrixEigen.jordan_formsDJJ6JJr:c \V3/VB#r<)r-r6rs&,r7left_eigenvectsMatrixEigen.left_eigenvectss...r:c\V4#r<)r.r=s&r7singular_valuesMatrixEigen.singular_valuess %%r:rtNTr)FFF)%rxryrzr{r|rrrrrr r rrrrrrr"r&r)r!r"r%r&r'r(r)r*r+r,r-r.r#r$r}r~rs@r7rrZstV040I%1<++//++//$$K/&*4););I)4)<))F)F )B)J)J$)>)F)F )B)J)J$)7)?)?M)5)=)=K)9)A)AO)9)A)AO)7)?)?M)B)J)J$$r:rcJa]tRtRtoRtRR/RltRtRtRtR t R t Vt R #) MatrixCalculusiz,Provides calculus-related matrix operations.evaluateTc ^RIHpV!V.VO5RV/p\V\4'gV'dVP 4#V#)zCalculate the derivative of each element in the matrix. Examples ======== >>> from sympy import Matrix >>> from sympy.abc import x, y >>> M = Matrix([[x, y], [1, 0]]) >>> M.diff(x) Matrix([ [1, 0], [0, 0]]) See Also ======== integrate limit )ArrayDerivativer.)$sympy.tensor.array.array_derivativesr0 isinstancer as_mutable)r6r.argsrr0derivs&$*, r7diffMatrixCalculus.diffs?* I?t?h?$&&8##% % r:c.aVPV3Rl4#)c&<VPS4#r<r6)rYargs&r71MatrixCalculus._eval_derivative..s s r: applyfunc)r6r;s&fr7_eval_derivativeMatrixCalculus._eval_derivatives~~344r:c 2aaVPVV3Rl4#)afIntegrate each element of the matrix. ``args`` will be passed to the ``integrate`` function. Examples ======== >>> from sympy import Matrix >>> from sympy.abc import x, y >>> M = Matrix([[x, y], [1, 0]]) >>> M.integrate((x, )) Matrix([ [x**2/2, x*y], [ x, 0]]) >>> M.integrate((x, 0, 2)) Matrix([ [2, 2*y], [2, 0]]) See Also ======== limit diff c(<VP!S/SB#r<) integrate)rYr4rs&r7r<*MatrixCalculus.integrate..s T(DV(Dr:r>)r6r4rs&jlr7rDMatrixCalculus.integrates2~~DEEr:c aa\S\4'gSPS4oSP^,^8XdSP^,pM7SP^,^8XdSP^,pM \ R4hSP^,^8XdSP^,pM7SP^,^8XdSP^,pM \ R4hSPW#VV3Rl4#)aCalculates the Jacobian matrix (derivative of a vector-valued function). Parameters ========== ``self`` : vector of expressions representing functions f_i(x_1, ..., x_n). X : set of x_i's in order, it can be a list or a Matrix Both ``self`` and X can be a row or a column matrix in any order (i.e., jacobian() should always work). Examples ======== >>> from sympy import sin, cos, Matrix >>> from sympy.abc import rho, phi >>> X = Matrix([rho*cos(phi), rho*sin(phi), rho**2]) >>> Y = Matrix([rho, phi]) >>> X.jacobian(Y) Matrix([ [cos(phi), -rho*sin(phi)], [sin(phi), rho*cos(phi)], [ 2*rho, 0]]) >>> X = Matrix([rho*cos(phi), rho*sin(phi)]) >>> X.jacobian(Y) Matrix([ [cos(phi), -rho*sin(phi)], [sin(phi), rho*cos(phi)]]) See Also ======== hessian wronskian z)``self`` must be a row or a column matrixz"X must be a row or a column matrixcB<SV,PSV,4#r<r:)rar`Xr6s&&r7r<)MatrixCalculus.jacobian..sDGLL1,>r:)r2r/rshape TypeError)r6rImnsff r7jacobianMatrixCalculus.jacobiansH!Z(( ! A ::a=A  1 A ZZ]a  1 AGH H 771:? A WWQZ1_ A@A Ayy>??r:c .aVPV3Rl4#)a&Calculate the limit of each element in the matrix. ``args`` will be passed to the ``limit`` function. Examples ======== >>> from sympy import Matrix >>> from sympy.abc import x, y >>> M = Matrix([[x, y], [1, 0]]) >>> M.limit(x, 2) Matrix([ [2, y], [1, 0]]) See Also ======== integrate diff c$<VP!S!#r<)limit)rYr4s&r7r<&MatrixCalculus.limit..+s r:r>)r6r4s&jr7rSMatrixCalculus.limits*~~677r:rtN) rxryrzr{r|r6r@rDrOrSr}r~rs@r7r-r-s06485F67@r88r:r-ca]tRtRtoRt]!R4]3RltRtRt Rt Rt RR lt R t R tR tRR ltRRltRtRtRtRtVtR#)MatrixDeprecatedi/z+A class to house deprecated matrix methods.rWc&VPVR7#))rY)r\r[s&&&r7berkowitz_charpoly#MatrixDeprecated.berkowitz_charpoly1s}}q}!!r:c &VPRR7#)zOComputes determinant using Berkowitz method. See Also ======== det berkowitz rvrRrhr=s&r7 berkowitz_detMatrixDeprecated.berkowitz_det4sxx{x++r:c &VP!R/VB#)zWComputes eigenvalues of a Matrix using Berkowitz method. See Also ======== berkowitz rt)rr%s&,r7berkowitz_eigenvals$MatrixDeprecated.berkowitz_eigenvals?s~~&&&r:c VP.r!VP4F$pVPWR,,4V)pK& \V4#)zPComputes principal minors using Berkowitz method. See Also ======== berkowitz )onervappendtuple)r6signminorspolys& r7berkowitz_minors!MatrixDeprecated.berkowitz_minorsIsExxfNN$D MM$b/ *5D%V}r:c^RIHpRpV'gV#VP'g \4hYPrC^.V^, ,p\ V^R4FpV!V^,V4V^, rW8RV13,)VRV1V3,rVRV1RV13,W8V3,)rV .p \ ^V^, 4F!p V P W&/*=<sympy.core.basicrsympy.core.symbolrcommonr exceptionsr utilitiesrrr determinantr r r r rrrrrrrrrrr reductionsrrrr subspacesrrrr eigenr!r"r#r$r%r&r'r(r)r*r+r,r-r. matrixbaser/__doctest_requires__r2rrrr-rWrtr:r7rs ## ,DD A@JJ6666#-0