Otj< ddlmZmZddlmZddlmZmZmZddl m Z m Z ddl m Z mZddlmZddlmZddlmZmZmZmZmZmZmZdd lmZdd lmZdd lm Z dd l!m"Z#d dl$m%Z%d dl&m'Z'd dl(m)Z)d dl*m*Z*d dl+m,Z,d dl-m.Z.m/Z/m0Z0m1Z1Gdde'e Z2e j3e e2fe2dZ4dZ5dZ6dZ7dZ8dZ9dZ:dZ;dZd Z?d!Z@e@ed<d"S)#)askQ) handlers_dict)BasicsympifyS)mulMul)NumberIntegerDummyadjoint)rm_idunpacktypedflattenexhaustdo_onenew)NonInvertibleMatrixError) MatrixBase)sympy_deprecation_warning)validate_matmul_integer)Inverse) MatrixExpr)MatPow transpose)PermutationMatrix) ZeroMatrixIdentityGenericIdentity OneMatrixceZdZdZdZeZdddddZedZ e dZ dd Z d Z d Zfd Zd ZdZdZdZdZdZddZdZxZS)MatMula A product of matrix expressions Examples ======== >>> from sympy import MatMul, MatrixSymbol >>> A = MatrixSymbol('A', 5, 4) >>> B = MatrixSymbol('B', 4, 3) >>> C = MatrixSymbol('C', 3, 6) >>> MatMul(A, B, C) A*B*C TFN)evaluatecheck_sympifycl|sjSttfd|}|r"ttt|}t jg|R}|\}}|tddd|dur t||s|S|r |S|S)Ncj|kSN)identity)iclss _/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/sympy/matrices/expressions/matmul.pyz MatMul.__new__..0sS\Q%6zaPassing check to MatMul is deprecated and the check argument will be removed in a future version.z1.11z,remove-check-argument-from-matrix-operations)deprecated_since_versionactive_deprecations_targetF) r/listfiltermaprr__new__as_coeff_matricesrvalidate _evaluate)r1r)r*r+argsobjfactormatricess` r2r:zMatMul.__new__*s < F6666==>>  ,GT**++DmC'$'''0022   %s)/+Y [ [ [ [    h   M  &==%% % r4c t|Sr.) canonicalize)r1exprs r2r=zMatMul._evaluateJsD!!!r4cXd|jD}|dj|djfS)Nc g|] }|j | S is_Matrix.0args r2 z MatMul.shape..Ps>>>C >C>>>r4r)r>rowscols)selfrAs r2shapez MatMul.shapeNs0>>49>>>  (2,"344r4c t ddlm}ddlm |\}}t |dkr||d||fzSdgt |dzzdgt |dz z}|d<|d<d} |d| tdt |D]}t|<t|ddD]\}} | j ddz ||<fdt|D}tj |} t fd |Drd }||| gtdddgt |z|Rz} td |Dsd }|r| n| S) Nr)Sum)ImmutableMatrixrrNc3@Kd} td|zV|dz })NrTzi_%ir )counters r2fzMatMul._entry..fbs7G FW,-----1  r4dummy_generatorcdg|],\}}|||dz-S)r)rY)_entry)rKr0rLrYindicess r2rMz!MatMul._entry..os>|||^d^_adCJJwqz71Q3<JYY|||r4c3BK|]}|VdSr.has)rKvrUs r2 z MatMul._entry..qs/88!quu_%%888888r4Tc3NK|] }t|ttfV!dSr.) isinstancer int)rKr`s r2raz MatMul._entry..ys0EEQ:a'300EEEEEEr4F)sympy.concrete.summationsrTsympy.matrices.immutablerUr;lengetrangenext enumeraterRr fromiteranyzipdoit)rQr0jexpandkwargsrTcoeffrA ind_rangesrXrL expr_in_sumresultrUrYr\s @@@r2r[z MatMul._entrySs"111111<<<<<<0022x x==A  8A;q!t,, ,&#h--!+,VS]]Q./       !**%6<<q#h--(( / /Ao..GAJJ" .. - -FAsIaL1,JqMM|||||hqrzh{h{|||l8,, 8888x888 8 8 FssWQrT]QCJ$7DD EE*EEEEE F &2v{{}}}F2r4cd|jD}d|jD}t|}|jdurtd||fS)Nc g|] }|j | SrGrHrKxs r2rMz,MatMul.as_coeff_matrices..~s;;;q{;1;;;r4c g|] }|j | SrGrHrys r2rMz,MatMul.as_coeff_matrices..s888!AK8A888r4Fz3noncommutative scalars in MatMul are not supported.)r>r is_commutativeNotImplementedError)rQscalarsrArss r2r;zMatMul.as_coeff_matrices}s\;;di;;;88ty888W   5 ( (%&[\\ \hr4cF|\}}|t|fSr.)r;r(rQrsrAs r2 as_coeff_mmulzMatMul.as_coeff_mmuls'0022xfh'''r4c ntt|jdi|}||SNrG)superr(rqr=)rQrrexpanded __class__s r2rqz MatMul.expands7-5&&-7777~~h'''r4c|\}}t|gd|dddDRS)aTransposition 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 c,g|]}t|SrGr rJs r2rMz*MatMul._eval_transpose..s>>>Ys^^>>>r4NrN)r;r(rors r2_eval_transposezMatMul._eval_transposesa&0022x @>>x"~>>>@@@@D Gr4chtd|jdddDS)Nc,g|]}t|SrGrrJs r2rMz(MatMul._eval_adjoint..s@@@ @@@r4rN)r(r>rorQs r2 _eval_adjointzMatMul._eval_adjoints4@@ $$B$@@@AFFHHHr4c|\}}|dkr&ddlm}|||zSdS)Nr)trace)rrro)rQr@mmulrs r2 _eval_tracezMatMul._eval_tracesV))++  Q;; $ $ $ $ $ $EE$))++... . ;r4c ddlm}|\}}t|}||jzt t t||zS)Nr) Determinant)&sympy.matrices.expressions.determinantrr; only_squaresrOr r7r9)rQrr@rAsquare_matricess r2_eval_determinantzMatMul._eval_determinants]FFFFFF1133&1ty 3So-N-N(O(O#PPPr4ctd|jDr3td|jdddDSt |S)Nc3NK|] }t|t|jV!dSr.)rcr is_squarerJs r2raz'MatMul._eval_inverse..s3QQZZ5P5PQs}QQQQQQr4c3rK|]2}t|tr|n|dzV3dS)rNN)rcrinverserJs r2raz'MatMul._eval_inverse..sU",C!r(rorrs r2 _eval_inversezMatMul._eval_inversesq QQ QQQ Q Q #y2dff   t}}r4c dd}|r!tfd|jD}n|j}tt |}|S)NdeepTc32K|]}|jdiVdSr)ro)rKrLhintss r2razMatMul.doit..s5@@s**E**@@@@@@r4)rhtupler>rCr()rQrrr>rDs ` r2roz MatMul.doitscyy&&  @@@@di@@@@@DD9DFDM** r4c djD}djD}|rSt|}t|}|r3|r1t||krtdfd|Dz||gS)Nc g|] }|j | SrGr|rys r2rMz#MatMul.args_cnc..s <<<1+;<1<<.s AAA!0@AAAAAr4z"repeated commutative arguments: %scjg|]/}tj|dk-|0Sr)r7r>count)rKcirQs r2rMz#MatMul.args_cnc..s;!X!X!X$ty//:O:OPR:S:SVW:W:W":W:W:Wr4)r>rgset ValueError)rQcsetwarnrrcoeff_ccoeff_ncclens` r2args_cnczMatMul.args_cncs<.s,IIIFAscggajjIaIIIr4c\g|](}|jr|n|)SrG)rIro)rKr0rs r2rMz8MatMul._eval_derivative_matrix_lines..s;+s+s+sZ[1;,UIIaLL,=,=,?,?,?TU+s+s+sr4r) r!rrkr>r(rlr$rRreversed_eval_derivative_matrix_lines append_first append_secondappend) rQrz with_x_indlinesind left_args right_args right_matleft_revdr0rs ` @r2rz$MatMul._eval_derivative_matrix_linessT((((((IIIIi &:&:III   C $3$I3q566*J 4"OOJ77 $TZ]33  3!??+s+s+s+s_ghq_r_r+s+s+stt#DJqM22 #<s r2newmulrs2 Aw!||ABBx v    r4ctd|jDr7d|jD}t|dj|djS|S)Nc3@K|]}|jp |jo|jVdSr.)is_zerorI is_ZeroMatrixrJs r2razany_zeros..sG , , ; ?3=>S-> , , , , , ,r4c g|] }|j | SrGrHrJs r2rMzany_zeros..s===Cs}=C===r4rrN)rmr>r#rOrP)r rAs r2 any_zerosrsj  , ,"%( , , ,,,?==38===(1+*HRL,=>>> Jr4cjtd|jDs|Sg}|jd}|jddD]W}t|ttfr"t|ttfr||z}@|||}X||t |S)a Merge 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] c3@K|]}t|tVdSr.)rcrrJs r2raz!merge_explicit..s,BBsz#z**BBBBBBr4rrN)rmr>rcrr rr()matmulnewargslastrLs r2merge_explicitrs8 BBfkBBB B B G ;q>D{122 cJ/ 0 0 ZzSYFZ5[5[ #:DD NN4 DD NN4 7 r4c|j\}}td|}||krt|g|jRS|S)z Remove 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 c|jduS)NT) is_Identityrzs r2r3zremove_ids..6s Q]d2r4)rrrr>)r r@rrvs r2 remove_idsr'sY%3$&&LFD 3U22 3 3D 9 9F ~~f+v{++++ r4cP|j\}}|dkr t|g|RS|SNr)r;r)r r@rAs r2factor_in_frontr<s;,s,..FH {{f(x(((( Jr4c|j\}}|dg}tdt|D]}|d}||}t|tryt|jt r_|jj}t|}t||| dkr(|d| t|j dgz}t|trt|jt ru|jj} t| }t| ||||zkrr7r$rRrrrrOnerorrrr)r r@r>new_argsr0ABBargslAargsr/rpA_baseA_expB_baseB_expnew_exp B_base_invs r2combine_powersrBs)3(**LFDQyH 1c$ii 00 RL G a ! ! j&?&? EJEE AE{{hrssm++#CaRC=HQWQZ,@,@+AA a ! ! j&?&? EJEE AE{{d1QqS5k))#AGAJ//' q!A#''A&DGG ;%  1;%#7#7 OOA     a  %FMFEEquEF a  %FMFEEquEF V  emG!&'2277U7CCHRL FJ//  "#^^-- + " " "!  "%&J*>*>%-%fg66;;;GG  & $8 $ $ $$sI&& I54I5c|j}t|}|dkr|S|dg}td|D]}|d}||}t|trEt|tr0|jd}|jd}t ||z|d<l||t |S)zGRefine products of permutation matrices as the products of cycles. rrrN)r>rgrircr"rr() r r>rrvr0rrcycle_1cycle_2s r2combine_permutationsrs 8D D A1uu 1gYF 1a[[   2J G a* + +  q+ , , fQiGfQiG*7W+<==F2JJ MM!     6?r4c|j\}}|dg}|ddD]}|d}t|trt|ts||J||t|jd|jd||jdz}t |g|RS)zj Combine products of OneMatrix e.g. OneMatrix(2, 3) * OneMatrix(3, 4) -> 3 * OneMatrix(2, 4) rrNrN)r;rcr&rpoprRr)r r@r>rrrs r2combine_one_matricesrs )3(**LFDQyH !""X RL!Y'' z!Y/G/G  OOA      !'!*agaj99:::!'!* & $8 $ $ $$r4c |jtdkrrddlm}djr)djr|fddjDSdjr)djr|fddjDS|S)zr Simplify MatMul expressions but distributing rational term to MatMul. e.g. 2*(A+B) -> 2*A + 2*B rr)MatAddrc`g|]*}t|d+Srr(rorKmatr>s r2rMz$distribute_monom..s3PPPCF3Q005577PPPr4c`g|]*}td|+S)rrrs r2rMz$distribute_monom..s3PPPCF47C005577PPPr4)r>rgmataddr is_MatAdd is_Rational)r rr>s @r2distribute_monomrs 8D 4yyA~~"""""" 7  Ra!4 R6PPPP47<PPPQ Q 7  Ra!4 R6PPPP47<PPPQ Q Jr4c|dkSrrGrs r2r3r3s klpqkqr4c 6|dj|djkrtdg}d}t|D]Y\}}|j||jkr>|t |||dz|dz}Z|S)z'factor matrices only if they are squarerrNz!Invalid matrices being multipliedr)rOrP RuntimeErrorrkrr(ro)rAoutstartr0Ms r2rrs{8B<,,,>??? C E(##1 6Xe_) ) ) JJvxac 2388:: ; ; ;aCE Jr4cPg}g}|jD]4}|jr||||5|d}|ddD]}||jkr=t t j||rt|jd}J|| kr=t t j ||rt|jd}|||}||t|S)z >>> from sympy import MatrixSymbol, Q, assuming, refine >>> X = MatrixSymbol('X', 2, 2) >>> expr = X * X.T >>> print(expr) X*X.T >>> with assuming(Q.orthogonal(X)): ... print(refine(expr)) I rrN) r>rIrTrr orthogonalr$rR conjugateunitaryr()rD assumptionsrexprargsr>rrLs r2 refine_MatMulrs"GH !! > ! OOD ! ! ! ! NN4 A;D| $&==Sc!2!2K@@=CIaL))DD DNN$$ $ $QYs^^[)I)I $CIaL))DD NN4 DD NN4 7 r4N)Asympy.assumptions.askrrsympy.assumptions.refiner sympy.corerrrsympy.core.mulr r sympy.core.numbersr r sympy.core.symbolrsympy.functionsrsympy.strategiesrrrrrrrsympy.matrices.exceptionsrsympy.matrices.matrixbasersympy.utilities.exceptionsr!sympy.matrices.expressions._shaperr<rrmatexprrmatpowrr! permutationr"specialr#r$r%r&r(register_handlerclassrrrrrrrrrrulesrCrrrGr4r2r%s&((((((((222222((((((((((########........############>>>>>>000000@@@@@@QQQQQQ ******EEEEEEEEEEEESSSSSZSSSj3-000 (((T* =%=%=%~,%%%("i-A>SY[`[`aqaq[r[rOW.B Dwuufffen56677    D( hr4