\h< SSKJrJr SSKJr SSKJrJrJr SSK J r J r SSK J r Jr SSKJr SSKJr SSKJrJrJrJrJrJrJr SS KJr SS KJr SS KJ r SS K!J"r# S SK$J%r% S SK&J'r' S SK(J)r) S SK*J*r* S SK+J,r, S SK-J.r.J/r/J0r0J1r1 "SS\'\ 5r2\ Rf"\ \24\25 Sr4Sr5Sr6Sr7Sr8Sr9Sr:Sr;Sr<\<\5\7\;\9\\"S5\6\8\\:4 r=\"\"\2\"\=6055r>S r?S!r@\@\S'g")#)askQ) handlers_dict)BasicsympifyS)mulMul)NumberIntegerDummy)adjoint)rm_idunpacktypedflattenexhaustdo_onenew)NonInvertibleMatrixError) MatrixBase)sympy_deprecation_warning)validate_matmul_integer)Inverse) MatrixExpr)MatPow) transpose)PermutationMatrix) ZeroMatrixIdentityGenericIdentity OneMatrixc^\rSrSrSrSr\"5rSSSS.Sjr\ S5r \ S 5r SS jr S rS rU4S jrSrSrSrSrSrSrSSjrSrSrU=r$)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 TFN)evaluatecheck_sympifycj^U(d TR$[[U4SjU55nU(a[[[U55n[ R "T/UQ76nUR5upgUb [SSSS9 USLa[U6 U(dU$U(aTRU5$U$)Nc">TRU:g$N)identity)iclss Y/var/www/auris/envauris/lib/python3.13/site-packages/sympy/matrices/expressions/matmul.py MatMul.__new__..0sS\\Q%6zaPassing check to MatMul is deprecated and the check argument will be removed in a future version.z1.11z,remove-check-argument-from-matrix-operations)deprecated_since_versionactive_deprecations_targetF) r.listfiltermaprr__new__as_coeff_matricesrvalidate _evaluate)r0r(r)r*argsobjfactormatricess` r1r:MatMul.__new__*s<< F6=> GT*+DmmC'$'002   %s)/+Y [   h M ==% % r4c[U5$r-) canonicalize)r0exprs r1r=MatMul._evaluateJs D!!r4cURVs/sHoR(dMUPM nnUSRUSR4$s snf)Nr)r> is_Matrixrowscols)selfargrAs r1shape MatMul.shapeNsB#'99>9C C9>   (2,"3"344?s A A c 2^SSKJn SSKJm UR 5upg[ U5S:Xa XgSX4-$S/[ U5S--nS/[ U5S- -n XS'X(S'Sn UR SU "55n [S[ U55Hn[U 5X'M [USS5HupU RSS- X'M [U5VV s/sHupU RXXS-U S9PM nnn [R"U5n [U4S jU55(aS nXe"U /[USSS/[ U 5-U 5Q76-n[S U 55(dS nU(aUR!5$U$s sn nf) Nr)Sum)ImmutableMatrixrrHc3># Sn[SU-5v US- nM7f)Nrzi_%ir )counters r1fMatMul._entry..fbs+GFW,--1 sdummy_generator)rWc3D># UHoRT5v M g7fr-)has).0vrRs r1 MatMul._entry..qs8x!uu_%%xs Tc3N# UHn[U[[45v M g7fr-) isinstancer int)rZr[s r1r\r]ysE*Q:a'300*s#%F)sympy.concrete.summationsrQsympy.matrices.immutablerRr;lengetrangenext enumeraterN_entryr fromiteranyzipdoit)rLr/jexpandkwargsrQcoeffrAindices ind_rangesrUrWrM expr_in_sumresultrRs @r1rh MatMul._entrySs1<002 x=A A;qt,, ,&#h-!+,VS]Q./     !**%6<q#h-(Ao.GJ) " .FAIIaL1,JM/hqrzh{|h{^d^_CJJwz7Q3<JYh{|ll8, 8x8 8 8FsWQr]QCJ$7D E*EEEF &v{{}2F2}s"$FcURVs/sHoR(aMUPM nnURVs/sHoR(dMUPM nn[U6nURSLa [ S5eXC4$s snfs snf)NFz3noncommutative scalars in MatMul are not supported.)r>rIr is_commutativeNotImplementedError)rLxscalarsrArps r1r;MatMul.as_coeff_matrices}sm"ii;i{{1i;#yy8y!KKAy8W    5 (%&[\ \ <8sBBBBc:UR5upU[U64$r-)r;r&)rLrprAs r1 as_coeff_mmulMatMul.as_coeff_mmuls"002fh'''r4c N>[[U] "S0UD6nURU5$N)superr&rnr=)rLroexpanded __class__s r1rn MatMul.expands&-77~~h''r4c UR5up[U/USSS2Vs/sHn[U5PM snQ76R5$s snf)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 NrH)r;r&rrl)rLrprArMs r1_eval_transposeMatMul._eval_transposesT&002 @/7"~>~Ys^~>@@D G>sA c[URSSS2Vs/sHn[U5PM sn6R5$s snf)NrH)r&r>rrl)rLrMs r1 _eval_adjointMatMul._eval_adjoints8 $B$@ @AFFHH@sAcpUR5upUS:waSSKJn X"UR55-$g)Nr)trace)r}rrl)rLr@mmulrs r1 _eval_traceMatMul._eval_traces7))+  Q; $E$))+.. . r4c SSKJn UR5up#[U6nX R-[ [ [X556-$)Nr) Determinant)&sympy.matrices.expressions.determinantrr; only_squaresrJr r7r9)rLrr@rAsquare_matricess r1_eval_determinantMatMul._eval_determinantsAF113&1yy 3S-N(O#PPPr4c[SUR55(a-[SURSSS256R5$[ U5$)Nc3h# UH(n[U[5(dMURv M* g7fr-)r_r is_squarerZrMs r1r\'MatMul._eval_inverse..sQ ZZ5P}s}} s22c3v# UH/n[U[5(aUR5OUS-v M1 g7f)rHN)r_rinversers r1r\rs2.",C!r&rlr)rLs r1 _eval_inverseMatMul._eval_inversesR Q Q Q Q#yy2df   t}r4c ^TRSS5nU(a [U4SjUR55nO URn[[ U65nU$)NdeepTc3F># UHoR"S0TD6v M g7fr)rl)rZrMhintss r1r\MatMul.doit..s@is*E*is!)rdtupler>rDr&)rLrrr>rEs ` r1rl MatMul.doitsHyy& @dii@@D99DFDM* r4c URVs/sHoDR(dMUPM nnURVs/sHoDR(aMUPM nnU(a|[U5n[U5nU(a_U(aX[U5U:waI[ SUVs/sH/n[ UR5R U5S:dM-UPM1 sn-5eXV/$s snfs snfs snf)Nz"repeated commutative arguments: %sr)r>rwrcset ValueErrorr7count) rLcsetwarnrorycoeff_ccoeff_ncclencis r1args_cncMatMul.args_cncs"iirYr&rir"rNreversedrIrl_eval_derivative_matrix_lines append_first append_secondappend) rLryrr/rM with_x_indlinesind left_args right_args right_matleft_revds r1r$MatMul._eval_derivative_matrix_liness0(&/ &:I&:FAcggaja&: IC $3I3q56*J"OOJ7 $TZZ]3 !??_ghq_r+s_rZ[;;IaL,=,=,?TU,U_r+st#DJJqM2 #<USS:XaUSSn[[/UQ76$)Nrr)rr&)r>s r1newmulrs( Aw!|ABx v  r4c[SUR55(aTURVs/sHoR(dMUPM nn[USRUSR 5$U$s snf)Nc3# UH7nUR=(d UR=(a URv M9 g7fr-)is_zerorI is_ZeroMatrixrs r1r\any_zeros..s2 ,"*3 ;; ?3==>S->-> ?"*s?ArrH)rjr>rIr!rJrK)r rMrAs r1 any_zerosrsg  ,"%(( ,,,#&88=8C}}C8=(1+**HRL,=,=>> J>s A8A8cf[SUR55(dU$/nURSnURSSHRn[U[[45(a![U[[45(aX#-nM?UR U5 UnMT UR U5 [ U6$)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# UHn[U[5v M g7fr-)r_rrs r1r\!merge_explicit..sBksz#z**ksrrN)rjr>r_rr rr&)matmulnewargslastrMs r1merge_explicitrs8 BfkkB B B G ;;q>D{{12 cJ/ 0 0ZzSYFZ5[5[:D NN4 D   NN4 7 r4cUR"5up[S5"U5nX2:wa[U/URQ76$U$)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 cURSL$)NT) is_Identityrys r1r2remove_ids..6s Q]]d2r4)r}rrr>)r r@rrts r1 remove_idsr'sB$$&LF 2 3D 9F ~f+v{{++ r4cPUR"5upUS:wa [U/UQ76$U$Nr)r;r)r r@rAs r1factor_in_frontr<s/,,.F {f(x(( Jr4cbUR"5upUS/n[S[U55GH_nUSnX$n[U[5(at[UR [ 5(aUUR Rn[U5n[U5X8*S:Xa"USU*[URS5/-nM[U[5(a[UR [ 5(alUR Rn [U 5n[U 5X$XH-:Xa8[URS5n XS'[XDU-5Hn XU 'M GM6URS:XdURS:XaURU5 GMj[U[5(aURupOU[Rp[U[5(aURupOU[RpX:Xa#X-n[U U5R!SS9US'GM[U["5(d=UR%5nUb)U U:Xa#X- n[U U5R!SS9US'GMNURU5 GMb [)U/UQ76$![&a SnN\f=f)zCombine consecutive powers with the same base into one, e.g. $$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 $$ rrrHNF)r)r;rercr_rrMr&r>r7r"rNrrrrOnerlrrrr)r r@r>new_argsr/ABBargslAargsr.rmA_baseA_expB_baseB_expnew_exp B_base_invs r1combine_powersrBsQ((*LFQyH 1c$i  RL G a ! !j&?&?EEJJEE AE{hrsm+#CaR=HQWWQZ,@+AA a ! !j&?&?EEJJEE AE{dQSk)#AGGAJ/' qA#A&G' ;;% 1;;%#7 OOA   a FFMFEquuE a FFMFEquuE  mG!&'277U7CHRL FJ// "#^^- %&J*>-%fg6;;;G a!d & $8 $$, "!  "sJ J.-J.cjURn[U5nUS:aU$US/n[SU5HwnUSnXn[U[5(aE[U[5(a0URSnURSn[ Xx-5US'MfUR U5 My [ U6$)zGRefine products of permutation matrices as the products of cycles. rrrH)r>rcrer_r rr&) r r>rrtr/rrcycle_1cycle_2s r1combine_permutationsrs 88D D A1u 1gYF 1a[ 2J G a* + + q+ , ,ffQiGffQiG*7+<=F2J MM!  6?r4cUR"5upUS/nUSSHnUSn[U[5(a[U[5(dURU5 MEUR 5 UR[UR SUR S55 XR S-nM [ U/UQ76$)z^ Combine products of OneMatrix e.g. OneMatrix(2, 3) * OneMatrix(3, 4) -> 3 * OneMatrix(2, 4) rrNrH)r;r_r$rpoprNr)r r@r>rrrs r1combine_one_matricesrs ((*LFQyH !"X RL!Y''z!Y/G/G OOA    !''!*aggaj9:''!* & $8 $$r4c URn[U5S:XaSSKJn USR(aQUSR (a=U"USRVs/sHn[ X1S5R5PM! sn6$USR(aRUSR (a>U"USRVs/sH n[ USU5R5PM" sn6$U$s snfs snf)zb Simplify MatMul expressions but distributing rational term to MatMul. e.g. 2*(A+B) -> 2*A + 2*B rr)MatAddr)r>rcmataddr is_MatAdd is_Rationalr&rl)r r>rmats r1distribute_monomrs 88D 4yA~" 7  a!4!447<<P?? C E(# 66X_)) ) JJvxac2388: ;aCE$ Jr4cT/n/nURH8nUR(aURU5 M'URU5 M: USnUSSHnXeR:Xa?[ [ R "U5U5(a[URS5nMQXeR5:Xa?[ [ R"U5U5(a[URS5nMURU5 UnM URU5 [U6$)z >>> from sympy import MatrixSymbol, Q, assuming, refine >>> X = MatrixSymbol('X', 2, 2) >>> expr = X * X.T >>> print(expr) X*X.T >>> with assuming(Q.orthogonal(X)): ... print(refine(expr)) I rrN) r>rIrTrr orthogonalr"rN conjugateunitaryr&)rE assumptionsrexprargsr>rrMs r1 refine_MatMulrsGH  >> OOD ! NN4  A;D| &&=Sc!2K@@CIIaL)D NN$ $QYYs^[)I)ICIIaL)D NN4 D NN4 7 r4N)Asympy.assumptions.askrrsympy.assumptions.refiner sympy.corerrrsympy.core.mulr r sympy.core.numbersr r sympy.core.symbolrsympy.functionsrsympy.strategiesrrrrrrrsympy.matrices.exceptionsrsympy.matrices.matrixbasersympy.utilities.exceptionsr!sympy.matrices.expressions._shaperr<rrmatexprrmatpowrr permutationr specialr!r"r#r$r&register_handlerclassrrrrrrrrrrulesrDrrrr4r1r%s(2((#.##>0@Q *EESZSj3-0 (T* =%~,%("i-A>SY[`aq[rOW.B Duffen567  D( hr4