+ i`6^RIHt^RIHtHt^RIHt^RIHt^RI H t ^RI H t ^RI Ht^RIHt^R IHtHt^R IHtHtHtHtHtHt^R IHtR t!R R]4tRt Rt!!RR]4t"R#))Counter)Mulsympify)Add) ExprBuilder)default_sort_key)log) MatrixExpr)validate_matadd_integer) ZeroMatrix OneMatrix)unpackflatten conditionexhaustrm_idsort)sympy_deprecation_warningcV'g \R4h\V4^8Xd V^,#\V!P4#)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 undefined) TypeErrorlenHadamardProductdoit)matricess*ڃ/Users/tonyclaw/.openclaw/workspace/skills/math-calculator/venv/lib/python3.14/site-packages/sympy/matrices/expressions/hadamard.pyhadamard_productrs=" =>> 8}{ H % * * ,,ctaa]tRt^)toRtRtRRRR/V3Rllt]R4tR t R t R t R t R t RtVtV;t#)ra 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`` TevaluateFcheckNc<\\\V44p\V4^8Xd \ R4h\ ;QJdRV4F 'dK RM RM !RV44'g \ R4hVe\RRRR 7VRJd \V!\SV`(!V.VO5!pV'dVPRR 7pV#) rz+HadamardProduct needs at least one argumentc3B"TFp\V\4xK R#5iN) isinstancer ).0args& r *HadamardProduct.__new__..Gs?$3:c:..$FTz 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_target)deep) listmaprr ValueErrorallrrvalidatesuper__new__r)clsrr argsobj __class__s&$$* rr3HadamardProduct.__new__AsC&' t9>JK Ks?$?sss?$???>? ?   %|)/+Y [   dOgoc)D) (((&C rc<VP^,P#r)r5shapeselfs&rr;HadamardProduct.shapeXsyy|!!!rc r\VPUu.uFqDP!W3/VBNK up!#uupir#)rr5_entry)r=ijkwargsr&s&&&, rr@HadamardProduct._entry\s/499E9CZZ//9EFFEs4cX^RIHp\\\ WP 44!#r) transpose)$sympy.matrices.expressions.transposerGrr-r.r5r=rGs& r_eval_transposeHadamardProduct._eval_transpose_sBSII%> ?@@rc aVP!V3RlVP4!p^RIHp^RIHpVPUu.uFp\ WS4'gKVNK ppV'dVPUu.uF qUV9gK VNK ppT!\V!Uu.uFp\P!V4NK up4P!VP!p\V.V,!p\V4#uupiuupiuupi)c3F<"TFqP!R/SBxK R#5i)N)r)r%rAhintss& rr''HadamardProduct.doit..ds>Iq66?E?Is!) MatrixBase)ImmutableMatrix)funcr5sympy.matrices.matrixbaserQsympy.matrices.immutablerRr$ziprfromiterreshaper;r canonicalize) r=rOexprrQrRrAexplicit remainderexpl_mats &l rrHadamardProduct.doitcsyy>DII>?8<#yyFy!Jq,EAAyF $(IICIq(1BIIC&),h()7A Q(w $H#hZ)%;=DD!!GC(sC2C26C7C7C<c&.p\VP4p\\V44FMpVRVW4,P V4.,W4^,R,pVP \ V!4KO \P!V4#r#) r-r5rangerdiffappendrrrW)r=xtermsr5rAfactorss&& r_eval_derivative HadamardProduct._eval_derivativesstDIIs4y!A2Ah$',,q/!22TA#$Z?G LL)73 4"||E""rc^RIHp^RIHp^RIHp\ VP 4UUu.uF wrVVPV4'gKVNK" ppp.pVEFp VP RV p VP V ^,Rp VP V ,PV4p \W,!p RR.p\ V4UUu.uF"wppVPV,^8wgK VNK$ pppV FpVPVP,pVPVP,p\V\V\VV.4V \VV.4.4.VO4pVP ^,P ^,P Vn^VnVP ^,P ^,P Vn^VnV.Vn VP'V4K EK V#uuppiuuppi)r ArrayDiagonalArrayTensorProduct _make_matrixN)r)0sympy.tensor.array.expressions.array_expressionsrjrl"sympy.matrices.expressions.matexprrn enumerater5has_eval_derivative_matrix_linesrr;_lines_first_line_index_second_line_indexr_first_pointer_parent_first_pointer_index_second_pointer_parent_second_pointer_indexrb)r=rcrjrlrnrAr& with_x_indlinesind left_args right_argsdhadamdiagonalrBel1l2subexprs&& rrw-HadamardProduct._eval_derivative_matrix_lines{sRWC&/ &:I&:FAcggajaa&: IC $3I3q56*J #<*>q*A*F*F')*&+2<<?+?+?+B+G+G(*+'#9 Q-@ EJQsG8 G8G>&G>rN)__name__ __module__ __qualname____firstlineno____doc__is_HadamardProductr3propertyr;r@rJrrfrw__static_attributes____classdictcell__ __classcell__r7 __classdict__s@@rrr)sY*U$.""GA" #''rrc\R\4p\V4pV!V4p\R\R44pV!V4pRp\RV4pV!V4p\ V\ 4'dn\ VP4p.pVP4F9wrgV^8XdVPV4KVP\Wg44K; \ V!p\R\\44pV!V4p\V4pV#)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"\V\4#r#r$rrcs&rcanonicalize.. jO4rc"\V\4#r#rrs&rrrrrc"\V\4#r#)r$r rs&rrrs Jq)4rc\;QJd&RVP4F 'gK RM RM!RVP44'd\VP!#V#)c3B"TFp\V\4xK R#5ir#)r$r )r%cs& rr'/canonicalize..absorb.. s9&Qz!Z((&r)TF)anyr5r r;rs&rabsorbcanonicalize..absorb s? 39!&&93339!&&9 9 9qww' 'Hrc"\V\4#r#rrs&rrrrrc"\V\4#r#rrs&rrr#rr)rrrrr$rrr5itemsrb HadamardPowerrrr)rcrulefunrtallynew_argbaseexps& rrYrYsf  4  D $-C AA  4 4 5 C AA  4  C AA!_%%IDaxt$}T78 ' W %  4 ! " C AA q A Hrc\V4p\V4pV^8XdV#VP'g W,#VP'd \R4h\W4#)z#cannot raise expression to a matrix)r is_Matrixr/r)rrs&&rhadamard_powerr-sQ 4=D #,C ax >>>y }}}>??  ##rc~aa]tRtRtoRtV3Rlt]R4t]R4t]R4t Rt Rt R t R t R tVtV;t#) 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<\V4p\V4pVP'dVP'd W,#\V\4'd"\V\4'd \ W4\ SV`WV4pV#r#)r is_scalarr$r r1r2r3)r4rrr6r7s&&& rr3HadamardPower.__new__rsdt}cl >>>cmmm;  dJ ' 'JsJ,G,G T goc- rc(VP^,#r:_argsr<s&rrHadamardPower.basezz!}rc(VP^,#)rrr<s&rrHadamardPower.exprrcVPP'dVPP#VPP#r#)rrr;rr<s&rr;HadamardPower.shapes. 99   99?? "xx~~rc VPpVPpVP'dVP!W3/VBpM/VP'dTpM\ RP V44hVP'dVP!W3/VBpWg,#VP'd TpWg,#\ RP V44h)z)The base {} must be a scalar or a matrix.z-The exponent {} must be a scalar or a matrix.)rrrr@rr/format)r=rArBrCrrabs&&&, rr@HadamardPower._entrysyyhh >>> A+F+A ^^^A;BB4HJ J === 1*6*Av ]]]A v ?FFsKM MrcZ^RIHp\V!VP4VP4#rF)rHrGrrrrIs& rrJHadamardPower._eval_transposesBYtyy1488<'C'CA $&'A ##$A yAH34 1Vs &F71F7rN)rrrrrr3rrrr;r@rJrfrwrrrrs@@rrr9se6p  ,=   rrN)# collectionsr sympy.corerrsympy.core.addrsympy.core.exprrsympy.core.sortingr&sympy.functions.elementary.exponentialr rtr !sympy.matrices.expressions._shaper r1"sympy.matrices.expressions.specialr r sympy.strategiesrrrrrrsympy.utilities.exceptionsrrrrYrrrNrrrs^#'/69QDA-0yjy|C L $WJWr