+ i< ^RIHtHt^RIHt^RIHtHtHt^RI H t H t ^RI H t Ht^RIHt^RIHt^RIHtHtHtHtHtHtHt^R IHt^R IHt^R IH t ^R I!H"t#^R I$H%t%^RI&H't'^RI(H)t)^RI*H*t*^RI+H,t,^RI-H.t.H/t/H0t0H1t1!RR]'] 4t2] Pf!] ]23]24Rt4Rt5Rt6Rt7Rt8Rt9Rt:Rt;Rt<]<]5]7];]9]]!R4]6]8]]:3 t=]!]!]2]!]=!/44t>Rt?R t@]@]R&R!#)")askQ) handlers_dict)BasicsympifyS)mulMul)NumberIntegerDummy)adjoint)rm_idunpacktypedflattenexhaustdo_onenew)NonInvertibleMatrixError) MatrixBase)sympy_deprecation_warning)validate_matmul_integer)Inverse) MatrixExpr)MatPow) transpose)PermutationMatrix) ZeroMatrixIdentityGenericIdentity OneMatrixcaa]tRt^toRtRt]!4tRRRRRR/Rlt] R 4t ] R 4t RR lt R tR tV3RltRtRtRtRtRtRtRRltRtRtVtV;t#)MatMulz A product of matrix expressions Examples ======== >>> from sympy import MatMul, MatrixSymbol >>> A = MatrixSymbol('A', 5, 4) >>> B = MatrixSymbol('B', 4, 3) >>> C = MatrixSymbol('C', 3, 6) >>> MatMul(A, B, C) A*B*C TevaluateFcheckN_sympifycaV'g SP#\\V3RlV44pV'd\\\V44p\ P !S.VO5!pVP4wrgVe\RRRR7VRJd \V!V'gV#V'dSPV4#V#)c"<SPV8g#N)identity)iclss&ځ/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/matrices/expressions/matmul.py MatMul.__new__..0sS\\Q%6zaPassing check to MatMul is deprecated and the check argument will be removed in a future version.z1.11z,remove-check-argument-from-matrix-operations)deprecated_since_versionactive_deprecations_targetF) r,listfiltermaprr__new__as_coeff_matricesrvalidate _evaluate)r.r&r'r(argsobjfactormatricessf$$$* r/r8MatMul.__new__*s<< F6=> GT*+DmmC'$'002   %s)/+Y [   h M ==% % r2c\V4#r+) canonicalize)r.exprs&&r/r;MatMul._evaluateJs D!!r2cVPUu.uFqP'gKVNK ppV^,PVR,P3#uupi)r)r< is_Matrixrowscols)selfargr?s& r/shape MatMul.shapeNsD#'99>9C CC9>   (2,"3"344?s AAc >a^RIHp^RIHoVP 4wrg\ V4^8XdWg^,W3,,#R.\ V4^,,pR.\ V4^, ,p W^&W(R &Rp VP RV !44p \^\ V44Fp\V 4W&K \VRR 4F!wrV P^,^, W&K# \V4UU u.uF,wrV PW,W^,,V R7NK. ppp \P!V4p \;QJdV3RlV4F 'gK RM R M!V3RlV44'dRpWe!V .\V^R ^.\ V 4,V 4O5!,p\;QJdR V 4F 'gK RM R M !R V 44'gR pV'dVP!4#T#uup pi) r)Sum)ImmutableMatrixNc3N"^p\RV,4xV^, pK 5i)zi_%ir )counters r/fMatMul._entry..fbs&GFW,--1 #%dummy_generator)rWc3D<"TFqPS4xK R#5ir+)has).0vrPs& r/ MatMul._entry..qs8x!uu_%%xs TFc3N"TFp\V\\34xK R#5ir+) isinstancer int)rZr[s& r/r\r]ysE*Q:a'300*rVrF)sympy.concrete.summationsrOsympy.matrices.immutablerPr9lengetrangenext enumeraterL_entryr fromiteranyzipdoit)rJr-jexpandkwargsrOcoeffr?indices ind_rangesrTrWrK expr_in_sumresultrPs&&&&, @r/rh MatMul._entrySs1<002 x=A A;qt,, ,&#h-!+,VS]Q./     !**%6<q#h-(Ao.GJ) " .FAIIaL1,JM/hqrzh{|h{^d^_CJJwz7Q3<JYh{|ll8, 38x83338x8 8 8FsWQr]QCJ$7D sE*EsssE*EEEF &v{{}2F2}s2HcVPUu.uFqP'dKVNK ppVPUu.uFqP'gKVNK pp\V!pVPRJd \ R4hWC3#uupiuupi)Fz3noncommutative scalars in MatMul are not supported.)r<rGr is_commutativeNotImplementedError)rJxscalarsr?rps& r/r9MatMul.as_coeff_matrices}sq"ii;i{{11i;#yy8y!KKAAy8W    5 (%&[\ \ <8sBBB B c<VP4wrV\V!3#r+)r9r%)rJrpr?s& r/ as_coeff_mmulMatMul.as_coeff_mmuls"002fh'''r2c N<\\V` !R/VBpVPV4#N)superr%rnr;)rJroexpanded __class__s&, r/rn MatMul.expands&-77~~h''r2c  VP4wr\V.VRRR1,Uu.uFp\V4NK upO5!P4#uupi)aPTransposition of matrix multiplication. Notes ===== The following rules are applied. Transposition for matrix multiplied with another matrix: `\left(A B\right)^{T} = B^{T} A^{T}` Transposition for matrix multiplied with scalar: `\left(c A\right)^{T} = c A^{T}` References ========== .. [1] https://en.wikipedia.org/wiki/Transpose NrF)r9r%rrl)rJrpr?rKs& r/_eval_transposeMatMul._eval_transposesT&002 @/7"~>~Ys^~>@@D G>sA c\VPRRR1,Uu.uFp\V4NK up!P4#uupi)NrF)r%r<rrl)rJrKs& r/ _eval_adjointMatMul._eval_adjoints8 $B$@ @AFFHH@sAc|VP4wrV^8wd#^RIHpW!VP44,#R#)rR)traceN)r}rrl)rJr>mmulrs& r/ _eval_traceMatMul._eval_traces7))+  Q; $E$))+.. . r2c ^RIHpVP4wr#\V!pW P,\ \ \W44!,#)r) Determinant)&sympy.matrices.expressions.determinantrr9 only_squaresrHr r5r7)rJrr>r?square_matricess& r/_eval_determinantMatMul._eval_determinantsAF113&1yy 3S-N(O#PPPr2c\;QJd&RVP4F 'dK RM RM!RVP44'd3\RVPRRR1,4!P4#\ V4#)c3j"TF)p\V\4'gKVPxK+ R#5ir+)r_r is_squarerZrKs& r/r\'MatMul._eval_inverse..sQ ZZ5P}s}} s33FTc3"TF4p\V\4'dVP4MVR,xK6 R#5i)rRNrF)r_rinversers& r/r\rs2.",C!NrF)allr<r%rlr)rJs&r/ _eval_inverseMatMul._eval_inversesf 3Q Q333Q Q Q Q#yy2df   t}r2c aSPRR4pV'dH\;QJd!.V3RlVP4FNK 5M!V3RlVP44pM VPp\\ V!4pV#)deepTc3F<"TFqP!R/SBxK R#5ir)rl)rZrKhintss& r/r\MatMul.doit..s@is*E*is!)rdtupler<rBr%)rJrrr<rCs&l r/rl MatMul.doitsZyy& 5@dii@55@dii@@D99DFDM* r2c VPUu.uFqDP'gKVNK ppVPUu.uFqDP'dKVNK ppV'd\V4p\V4pV'dfV'd^\V4V8wdN\ RVUu.uF0p\ VP4P V4^8gK.VNK2 up,4hWV.#uupiuupiuupi)z"repeated commutative arguments: %s)r<rwrcset ValueErrorr5count) rJcsetwarnrorycoeff_ccoeff_ncclencis &&&, r/args_cncMatMul.args_cncs"ii V#uuppiuupi)rR) TransposeN)rrrgr<rYr%rir!rLreversedrGrl_eval_derivative_matrix_lines append_first append_secondappend) rJryrr-rK with_x_indlinesind left_args right_args right_matleft_revds && r/r$MatMul._eval_derivative_matrix_liness4(&/ &:I&:FAcggajaa&: IC $3I3q56*J"OOJ7 $TZZ]3 !??_ghq_r+s_rZ[;;;IaL,=,=,?TU,U_r+st#DJJqM2 #<.s8 ,"*3 ;; ? ?3==>>S->-> ?"*sAAATFrF)rjr<rGr rHrI)r rKr?s& r/ any_zerosrs s ,"%(( ,sss ,"%(( ,,,#&88=8C}}CC8=(1+**HRL,=,=>> J>s B-5B-c\;QJd&RVP4F 'gK RM RM!RVP44'gV#.pVP^,pVPR,FXp\V\\34'd'\V\\34'd W#,pKEVP V4TpKZ VP V4\ V!#)a7Merge explicit MatrixBase arguments >>> from sympy import MatrixSymbol, Matrix, MatMul, pprint >>> from sympy.matrices.expressions.matmul import merge_explicit >>> A = MatrixSymbol('A', 2, 2) >>> B = Matrix([[1, 1], [1, 1]]) >>> C = Matrix([[1, 2], [3, 4]]) >>> X = MatMul(A, B, C) >>> pprint(X) [1 1] [1 2] A*[ ]*[ ] [1 1] [3 4] >>> pprint(merge_explicit(X)) [4 6] A*[ ] [4 6] >>> X = MatMul(B, A, C) >>> pprint(X) [1 1] [1 2] [ ]*A*[ ] [1 1] [3 4] >>> pprint(merge_explicit(X)) [1 1] [1 2] [ ]*A*[ ] [1 1] [3 4] c3B"TFp\V\4xK R#5ir+)r_rrs& r/r\!merge_explicit..sBksz#z**ksTFr)rjr<r_rr rr%)matmulnewargslastrKs& r/merge_explicitrs8 3BfkkB333BfkkB B B G ;;q>D{{2 cJ/ 0 0ZzSYFZ5[5[:D NN4 D   NN4 7 r2cVP!4wr\R4!V4pW28wd\V.VPO5!#V#)zRemove Identities from a MatMul This is a modified version of sympy.strategies.rm_id. This is necessary because MatMul may contain both MatrixExprs and Exprs as args. See Also ======== sympy.strategies.rm_id cVPRJ#r) is_Identityrys&r/r0remove_ids..6s Q]]d2r2)r}rrr<)r r>rrts& r/ remove_idsr'sB$$&LF 2 3D 9F ~f+v{{++ r2cTVP!4wrV^8wd\V.VO5!#V#rR)r9r)r r>r?s& r/factor_in_frontr<s/,,.F {f(x(( Jr2cVP!4wrV^,.p\^\V44EFpVR,pW$,p\V\4'd\VP \ 4'd_VP Pp\V4p\V4W8)R8Xd+VRV)\VP^,4.,pK\V\4'd\VP \ 4'dzVP Pp \V 4p\V 4W$WH,8XdA\VP^,4p WR&\WDV,4Fp WV &K EKXVPR8XgVPR8XdVPV4EK\V\4'dVPwrMT\Pr\V\4'dVPwrMT\PrW8Xd+W,p\V V4P!RR7VR&EK*\V\"4'gGVP%4pVe2V V8Xd+W, p\V V4P!RR7VR&EKVPV4EK \)V.VO5!# \&dRpLgi;i)zCombine consecutive powers with the same base into one, e.g. $$A \times A^2 \Rightarrow A^3$$ This also cancels out the possible matrix inverses using the knowledgebase of :class:`~.Inverse`, e.g., $$ Y \times X \times X^{-1} \Rightarrow Y $$ NF)rrF)r9rercr_rrKr%r<r5r!rLrrrrOnerlrrrr)r r>r<new_argsr-ABBargslAargsr,rmA_baseA_expB_baseB_expnew_exp B_base_invs& r/combine_powersrBsQ((*LFQyH 1c$i  RL G a ! !j&?&?EEJJEE AE{hrsm+#CaR=HQWWQZ,@+AA a ! !j&?&?EEJJEE AE{dQSk)#AGGAJ/' qA#A&G' ;;% 1;;%#7 OOA   a FFMFEquuE a FFMFEquuE  mG!&'277U7CHRL FJ// "#^^- %&J*>-%fg6;;;G a!d & $8 $$, "!  "s4K K,+K,cVPp\V4pV^8dV#V^,.p\^V4FpVR,pW,p\V\4'dS\V\4'd=VP^,pVP^,p\ Wx,4VR&K|VP V4K \ V!#)zGRefine products of permutation matrices as the products of cycles. rF)r<rcrer_rrr%) r r<rrtr-rrcycle_1cycle_2s & r/combine_permutationsrs 88D D A1u 1gYF 1a[ 2J G a* + + q+ , ,ffQiGffQiG*7+<=F2J MM!  6?r2cVP!4wrV^,.pVR,FpVR,p\V\4'd\V\4'gVPV4KKVP 4VP\VP ^,VP ^,44WP ^,,pK \ V.VO5!#)z^ Combine products of OneMatrix e.g. OneMatrix(2, 3) * OneMatrix(3, 4) -> 3 * OneMatrix(2, 4) rrF)r9r_r#rpoprLr)r r>r<rrrs& r/combine_one_matricesrs ((*LFQyH "XX RL!Y''z!Y/G/G OOA    !''!*aggaj9:''!* & $8 $$r2c <VPp\V4^8Xd^RIHpV^,P'd_V^,P 'dFT!V^,PUu.uF#p\ W1^,4P4NK% up!#V^,P'd`V^,P 'dGT!V^,PUu.uF$p\ V^,V4P4NK& up!#V#uupiuupi)zb Simplify MatMul expressions but distributing rational term to MatMul. e.g. 2*(A+B) -> 2*A + 2*B )MatAdd)r<rcmataddr is_MatAdd is_Rationalr%rl)r r<rmats& r/distribute_monomrs 88D 4yA~" 7   a!4!4!447<<P?? C E(# 66X_)) ) JJvxac2388: ;aCE$ Jr2c.p.pVPF9pVP'dVPV4K(VPV4K; V^,pVR,FpWeP8XdE\ \ P !V4V4'd\VP^,4pKWWeP48XdE\ \ P!V4V4'd\VP^,4pKVPV4TpK VPV4\V!#)z >>> from sympy import MatrixSymbol, Q, assuming, refine >>> X = MatrixSymbol('X', 2, 2) >>> expr = X * X.T >>> print(expr) X*X.T >>> with assuming(Q.orthogonal(X)): ... print(refine(expr)) I r) r<rGrTrr orthogonalr!rL conjugateunitaryr%)rC assumptionsrexprargsr<rrKs&& r/ refine_MatMulrsGH  >>> OOD ! NN4  A;D|| &&=Sc!2K@@CIIaL)D NN$ $QYYs^[)I)ICIIaL)D NN4 D NN4 7 r2N)Asympy.assumptions.askrrsympy.assumptions.refiner sympy.corerrrsympy.core.mulr r sympy.core.numbersr r sympy.core.symbolrsympy.functionsrsympy.strategiesrrrrrrrsympy.matrices.exceptionsrsympy.matrices.matrixbasersympy.utilities.exceptionsr!sympy.matrices.expressions._shaperr:rrmatexprrmatpowrr permutationrspecialr r!r"r#r%register_handlerclassrrrrrrrrrrulesrBrrrr2r/r(s(2((#.##>0@Q *EESZSj3-0 (T* =%~,%("i-A>SY[`aq[rOW.B Duffen567  D( hr2