\h`6SSKJr SSKJrJr SSKJr SSKJr SSK J r SSK J r SSK Jr SSKJr SS KJrJr SS KJrJrJrJrJrJr SS KJr S r"S S\5rSr Sr!"SS\5r"g))Counter)Mulsympify)Add) ExprBuilder)default_sort_key)log) MatrixExpr)validate_matadd_integer) ZeroMatrix OneMatrix)unpackflatten conditionexhaustrm_idsort)sympy_deprecation_warningczU(d [S5e[U5S:XaUS$[U6R5$)aA Return the elementwise (aka Hadamard) product of matrices. Examples ======== >>> from sympy import hadamard_product, MatrixSymbol >>> A = MatrixSymbol('A', 2, 3) >>> B = MatrixSymbol('B', 2, 3) >>> hadamard_product(A) A >>> hadamard_product(A, B) HadamardProduct(A, B) >>> hadamard_product(A, B)[0, 1] A[0, 1]*B[0, 1] z#Empty Hadamard product is undefinedr) TypeErrorlenHadamardProductdoit)matricess [/var/www/auris/envauris/lib/python3.13/site-packages/sympy/matrices/expressions/hadamard.pyhadamard_productrs=" =>> 8}{ H % * * ,,ch^\rSrSrSrSrSSS.U4Sjjr\S5rS r S r S r S r S r SrU=r$)r)a Elementwise product of matrix expressions Examples ======== Hadamard product for matrix symbols: >>> from sympy import hadamard_product, HadamardProduct, MatrixSymbol >>> A = MatrixSymbol('A', 5, 5) >>> B = MatrixSymbol('B', 5, 5) >>> isinstance(hadamard_product(A, B), HadamardProduct) True Notes ===== This is a symbolic object that simply stores its argument without evaluating it. To actually compute the product, use the function ``hadamard_product()`` or ``HadamardProduct.doit`` TFN)evaluatecheckc6>[[[U55n[U5S:Xa [ S5e[ SU55(d [ S5eUb [SSSS9 US La[U6 [TU]("U/UQ76nU(aURS S 9nU$) Nrz+HadamardProduct needs at least one argumentc3B# UHn[U[5v M g7fN) isinstancer ).0args r *HadamardProduct.__new__..Gs?$3:c:..$z Mix of Matrix and Scalar symbolszjPassing check to HadamardProduct 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)deep) listmaprr ValueErrorallrrvalidatesuper__new__r)clsr!r"argsobj __class__s rr5HadamardProduct.__new__AsC&' t9>JK K?$???>? ?   %|)/+Y [   dOgoc)D) (((&C rc4URSR$Nr)r7shapeselfs rr=HadamardProduct.shapeXsyy|!!!rc r[URVs/sHoDR"X40UD6PM sn6$s snfr%)rr7_entry)r?ijkwargsr(s rrBHadamardProduct._entry\s/499E9CZZ//9EFFEs4cVSSKJn [[[ XR 556$Nr) transpose)$sympy.matrices.expressions.transposerIrr/r0r7r?rIs r_eval_transposeHadamardProduct._eval_transpose_sBSII%> ?@@rc ^UR"U4SjUR56nSSKJn SSKJn URVs/sHn[ XS5(dMUPM nnU(ayURVs/sH oUU;dM UPM nnU"[U6Vs/sHn[R"U5PM sn5R"UR6n[U/U-6n[U5$s snfs snfs snf)Nc3F># UHoR"S0TD6v M g7f)N)r)r'rChintss rr)'HadamardProduct.doit..ds>Iq66?E?Is!r) MatrixBase)ImmutableMatrix)funcr7sympy.matrices.matrixbaserSsympy.matrices.immutablerTr&ziprfromiterreshaper=r canonicalize) r?rQexprrSrTrCexplicit remainderexpl_mats ` rrHadamardProduct.doitcsyy>DII>?8<#yyFy!Jq,EAyF $(IICIq(1BIIC&),h()7A Q(w $H#hZ)%;=DD!!GC(sC*C*4 C/C/ C4c/n[UR5n[[U55H<nUSUX4R U5/-X4S-S-nUR [ U65 M> [R"U5$Nr) r/r7rangerdiffappendrrrY)r?xtermsr7rCfactorss r_eval_derivative HadamardProduct._eval_derivativesstDIIs4y!A2Ah$',,q/!22TA#$Z?G LL)73 4"||E""rcSSKJn SSKJn SSKJn [ UR 5VVs/sHupVURU5(dMUPM! nnn/nUGHpn UR SU n UR U S-Sn UR U RU5n [X-6n SS/n[ U5VVs/sHunnURUS:wdMUPM nnnU HnURURnURURn[U[U[UU/5U [UU/5/5/UQ5nUR SR SR UlSUlUR SR SR UlSUlU/Ul UR'U5 M GMs U$s snnfs snnf) Nr ArrayDiagonalArrayTensorProduct _make_matrixr)rrr)0sympy.tensor.array.expressions.array_expressionsrmro"sympy.matrices.expressions.matexprrq enumerater7has_eval_derivative_matrix_linesrr=_lines_first_line_index_second_line_indexr_first_pointer_parent_first_pointer_index_second_pointer_parent_second_pointer_indexre)r?rfrmrorqrCr( with_x_indlinesind left_args right_argsdhadamdiagonalrDel1l2subexprs rrz-HadamardProduct._eval_derivative_matrix_lines{sRWC&/ &:I&:FAcggaja&: IC $3I3q56*J #<*>q*A*F*F')*&+2<<?+?+?+B+G+G(*+'#9 Q-@ EJQsG G7GGrP)__name__ __module__ __qualname____firstlineno____doc__is_HadamardProductr5propertyr=rBrLrrirz__static_attributes__ __classcell__r9s@rrr)sS*%*$.""GA" #''rrc[S[5n[U5nU"U5n[S[S55nU"U5nSn[SU5nU"U5n[ U[ 5(ak[ UR5n/nUR5H8upgUS:XaURU5 MUR[Xg55 M: [ U6n[S[[55nU"U5n[U5nU$)aCanonicalize the Hadamard product ``x`` with mathematical properties. Examples ======== >>> from sympy import MatrixSymbol, HadamardProduct >>> from sympy import OneMatrix, ZeroMatrix >>> from sympy.matrices.expressions.hadamard import canonicalize >>> from sympy import init_printing >>> init_printing(use_unicode=False) >>> A = MatrixSymbol('A', 2, 2) >>> B = MatrixSymbol('B', 2, 2) >>> C = MatrixSymbol('C', 2, 2) Hadamard product associativity: >>> X = HadamardProduct(A, HadamardProduct(B, C)) >>> X A.*(B.*C) >>> canonicalize(X) A.*B.*C Hadamard product commutativity: >>> X = HadamardProduct(A, B) >>> Y = HadamardProduct(B, A) >>> X A.*B >>> Y B.*A >>> canonicalize(X) A.*B >>> canonicalize(Y) A.*B Hadamard product identity: >>> X = HadamardProduct(A, OneMatrix(2, 2)) >>> X A.*1 >>> canonicalize(X) A Absorbing element of Hadamard product: >>> X = HadamardProduct(A, ZeroMatrix(2, 2)) >>> X A.*0 >>> canonicalize(X) 0 Rewriting to Hadamard Power >>> X = HadamardProduct(A, A, A) >>> X A.*A.*A >>> canonicalize(X) .3 A Notes ===== As the Hadamard product is associative, nested products can be flattened. The Hadamard product is commutative so that factors can be sorted for canonical form. A matrix of only ones is an identity for Hadamard product, so every matrices of only ones can be removed. Any zero matrix will make the whole product a zero matrix. Duplicate elements can be collected and rewritten as HadamardPower References ========== .. [1] https://en.wikipedia.org/wiki/Hadamard_product_(matrices) c"[U[5$r%r&rrfs rcanonicalize.. jO4rc"[U[5$r%rrs rrrrrc"[U[5$r%)r&r rs rrrs Jq)4rcl[SUR55(a[UR6$U$)Nc3B# UHn[U[5v M g7fr%)r&r )r'cs rr)/canonicalize..absorb.. s9&Qz!Z((&r+)anyr7r r=rs rabsorbcanonicalize..absorb s+ 9!&&9 9 9qww' 'Hrc"[U[5$r%rrs rrrrrrc"[U[5$r%rrs rrr#rr)rrrrr&rrr7itemsre HadamardPowerrrr)rfrulefunrtallynew_argbaseexps rr[r[sf  4  D $-C AA  4 4 5 C AA  4  C AA!_%%IDaxt$}T78 ' W %  4 ! " C AA q A Hrc[U5n[U5nUS:XaU$UR(dX-$UR(a [S5e[X5$)Nrz#cannot raise expression to a matrix)r is_Matrixr1r)rrs rhadamard_powerr-sM 4=D #,C ax >>y }}>??  ##rct^\rSrSrSrU4Sjr\S5r\S5r\S5r Sr Sr S r S r S rU=r$) ri9a Elementwise power of matrix expressions Parameters ========== base : scalar or matrix exp : scalar or matrix Notes ===== There are four definitions for the hadamard power which can be used. Let's consider `A, B` as `(m, n)` matrices, and `a, b` as scalars. Matrix raised to a scalar exponent: .. math:: A^{\circ b} = \begin{bmatrix} A_{0, 0}^b & A_{0, 1}^b & \cdots & A_{0, n-1}^b \\ A_{1, 0}^b & A_{1, 1}^b & \cdots & A_{1, n-1}^b \\ \vdots & \vdots & \ddots & \vdots \\ A_{m-1, 0}^b & A_{m-1, 1}^b & \cdots & A_{m-1, n-1}^b \end{bmatrix} Scalar raised to a matrix exponent: .. math:: a^{\circ B} = \begin{bmatrix} a^{B_{0, 0}} & a^{B_{0, 1}} & \cdots & a^{B_{0, n-1}} \\ a^{B_{1, 0}} & a^{B_{1, 1}} & \cdots & a^{B_{1, n-1}} \\ \vdots & \vdots & \ddots & \vdots \\ a^{B_{m-1, 0}} & a^{B_{m-1, 1}} & \cdots & a^{B_{m-1, n-1}} \end{bmatrix} Matrix raised to a matrix exponent: .. math:: A^{\circ B} = \begin{bmatrix} A_{0, 0}^{B_{0, 0}} & A_{0, 1}^{B_{0, 1}} & \cdots & A_{0, n-1}^{B_{0, n-1}} \\ A_{1, 0}^{B_{1, 0}} & A_{1, 1}^{B_{1, 1}} & \cdots & A_{1, n-1}^{B_{1, n-1}} \\ \vdots & \vdots & \ddots & \vdots \\ A_{m-1, 0}^{B_{m-1, 0}} & A_{m-1, 1}^{B_{m-1, 1}} & \cdots & A_{m-1, n-1}^{B_{m-1, n-1}} \end{bmatrix} Scalar raised to a scalar exponent: .. math:: a^{\circ b} = a^b c >[U5n[U5nUR(aUR(aX-$[U[5(a [U[5(a [ X5 [ TU]XU5nU$r%)r is_scalarr&r r3r4r5)r6rrr8r9s rr5HadamardPower.__new__rs`t}cl >>cmm;  dJ ' 'JsJ,G,G T goc- rc URS$r<_argsr>s rrHadamardPower.basezz!}rc URS$rbrr>s rrHadamardPower.exprrcURR(aURR$URR$r%)rrr=rr>s rr=HadamardPower.shapes+ 99  99?? "xx~~rc URnURnUR(aUR"X40UD6nO.UR(aUnO[ SR U55eUR(aUR"X40UD6nXg-$UR(aUnXg-$[ SR U55e)Nz)The base {} must be a scalar or a matrix.z-The exponent {} must be a scalar or a matrix.)rrrrBrr1format)r?rCrDrErrabs rrBHadamardPower._entrysyyhh >> A+F+A ^^A;BB4HJ J == 1*6*Av ]]A v ?FFsKM MrcZSSKJn [U"UR5UR5$rH)rJrIrrrrKs rrLHadamardPower._eval_transposesBYtyy1488<'C'CA $&'A ##$A yAH34 1Vs #F-FrP)rrrrrr5rrrr=rBrLrirzrrrs@rrr9sc6p  ,=   rrN)# collectionsr sympy.corerrsympy.core.addrsympy.core.exprrsympy.core.sortingr&sympy.functions.elementary.exponentialr rwr !sympy.matrices.expressions._shaper r3"sympy.matrices.expressions.specialr r sympy.strategiesrrrrrrsympy.utilities.exceptionsrrrr[rrrPrrrs^#'/69QDA-0yjy|C L $WJWr