a Юм>> 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 undefinedщrcSsg|]}|js|СqSй)Zis_Identityй┌.0┌irr·s/Users/vegardjervell/Documents/master/model/venv/lib/python3.9/site-packages/sympy/matrices/expressions/hadamard.py┌ $єz$hadamard_product..N)┌ TypeError┌validate┌len┌HadamardProduct┌doit)Zmatricesrrr┌hadamard_product srcs`eZdZdZdZdddЬЗfddД ZeddДГZd d ДZddДZ d dДZ ddДZddДZЗZ S)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`` TF)┌evaluate┌checkcsBttt|ГГ}|rt|ОtГj|g|вRО}|r>|jddН}|S)NF)┌deep)┌list┌maprr┌super┌__new__r)┌clsrr ┌args┌objй┌ __class__rrr%@szHadamardProduct.__new__cCs|jdjSйNr)r'┌shapeй┌selfrrrr,JszHadamardProduct.shapecstЗЗЗfddД|jDГОS)Ncs g|]}|jИИfiИдОСqSr)┌_entryйr┌argйr┌j┌kwargsrrrOrz*HadamardProduct._entry..)rr')r.rr3r4rr2rr/NszHadamardProduct._entrycCs ddlm}ttt||jГГОSйNr)┌ transpose)┌$sympy.matrices.expressions.transposer6rr"r#r'йr.r6rrr┌_eval_transposeQszHadamardProduct._eval_transposecsТ|jЗfddД|jDГО}ddlmЙddlm}ЗfddД|jDГЙИrКЗfddД|jDГ}|ddДtИОDГГj|jО}t |g|О}t |ГS) Ncsg|]}|jfiИдОСqSr)rr)┌ignoredrrrVrz(HadamardProduct.doit..rй┌ MatrixBase)┌ImmutableMatrixcsg|]}t|ИГr|СqSr)┌ isinstancerr;rrrZrcsg|]}|Иvr|СqSrrr)┌explicitrrr\rcSsg|]}tа|бСqSr)r┌fromiterrrrrr]s)┌funcr'Zsympy.matrices.matricesr<Zsympy.matrices.immutabler=┌zipZreshaper,r┌canonicalize)r.r:┌exprr=┌ remainderZexpl_matr)r<r?r:rrUs ■zHadamardProduct.doitcCsdg}t|jГ}tt|ГГD]>}|d|Е||а|бg||ddЕ}|аt|Обqtа|бSйNr) r"r'┌ranger┌diff┌appendrrr@)r.┌xZtermsr'rZfactorsrrr┌_eval_derivativeds ,z HadamardProduct._eval_derivativecs<ddlm}ddlm}ddlm}ЗfddДtИjГDГ}g}|D]Є}Иjd|Е}Иj|ddЕ} Иj|аИб} t| |О}dd g}Зfd dДt|ГDГ}| D]О} | j | j }| j | j}t|t|t||gГ|t||gГgГg|вГ}|jdjdj| _ d| _|jdjdj| _d| _|g| _ |а| бqжqD|S)Nrй┌ ArrayDiagonalй┌ArrayTensorProductй┌_make_matrixcsg|]\}}|аИбr|СqSr)Zhas)rrr1йrJrrrqrzAHadamardProduct._eval_derivative_matrix_lines..r)rщйщщcs"g|]\}}Иj|dkr|СqSйr)r,йrr3┌er-rrrzrrS)┌0sympy.tensor.array.expressions.array_expressionsrMrO┌"sympy.matrices.expressions.matexprrQ┌ enumerater'┌_eval_derivative_matrix_linesr┌_lines┌_first_line_index┌_second_line_indexr┌_first_pointer_parent┌_first_pointer_index┌_second_pointer_parent┌_second_pointer_indexrI)r.rJrMrOrQZ with_x_ind┌lines┌indZ left_argsZ right_args┌dZhadam┌diagonalr┌l1┌l2┌subexprr)r.rJrr]lsF ¤■ ў■z-HadamardProduct._eval_derivative_matrix_lines)┌__name__┌ __module__┌__qualname__┌__doc__Zis_HadamardProductr%┌propertyr,r/r9rrKr]┌ __classcell__rrr)rr(s rcGsTtddД|DГГstdГВ|d}|ddЕD] }|j|jkr.td||fГВq.dS)Ncss|]}|jVqdSйN)┌ is_Matrixr0rrr┌ Чrzvalidate..z Mix of Matrix and Scalar symbolsrrz"Matrices %s and %s are not aligned)┌allrr,r)r'┌A┌BrrrrЦsrc CsтtddДtГ}t|Г}||Г}tddДtddДГГ}||Г}ddД}tddД|Г}||Г}t|tГr╝dd lm}||jГ}g}|а бD],\}}|d krв|а |бqЖ|а t||ГбqЖt|О}tddДtt ГГ}||Г}t|Г}|S)aЄCanonicalize 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) cSs t|tГSrrйr>rrRrrr┌їrzcanonicalize..cSs t|tГSrrrxrRrrrry¤rcSs t|tГSrr)r>r rRrrrry■rcSs&tddД|jDГГrt|jОS|SdS)Ncss|]}t|tГVqdSrr)r>r )r┌crrrrtrz/canonicalize..absorb..)┌anyr'r r,rRrrr┌absorbs zcanonicalize..absorbcSs t|tГSrrrxrRrrrry rr)┌CounterrcSs t|tГSrrrxrRrrrryr)r rrrr>r┌collectionsr}r'┌itemsrI┌ HadamardPowerrrr) rJZruleZfunr|r}ZtallyZnew_arg┌base┌exprrrrCбsBS■ ■■ ■rCcCsBt|Г}t|Г}|dkr|S|js*||S|jr8tdГВt||ГS)Nrz#cannot raise expression to a matrix)rrs┌ ValueErrorrА)rБrВrrr┌hadamard_power(srДcsdeZdZdZЗfddДZeddДГZeddДГZedd ДГZd dДZ dd ДZ ddДZddДZЗZ S)rАaд 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 csdt|Г}t|Г}|jr$|jr$||S|jrP|jrP|j|jkrPtdа|j|jбГВtГа|||б}|S)NzGThe shape of the base {} and the shape of the exponent {} do not match.)r┌ is_scalarrsr,rГ┌formatr$r%)r&rБrВr(r)rrr%ms■ zHadamardPower.__new__cCs |jdSr+й┌_argsr-rrrrБ}szHadamardPower.basecCs |jdSrFrЗr-rrrrВБszHadamardPower.expcCs|jjr|jjS|jjSrr)rБrsr,rВr-rrrr,ЕszHadamardPower.shapecKsА|j}|j}|jr(|j||fi|дО}n|jr4|}ntdа|бГВ|jr^|j||fi|дО}n|jrj|}ntdа|бГВ||S)Nz)The base {} must be a scalar or a matrix.z-The exponent {} must be a scalar or a matrix.)rБrВrsr/rЕrГrЖ)r.rr3r4rБrВ┌a┌brrrr/Лs" zHadamardPower._entrycCsddlm}t||jГ|jГSr5)r7r6rАrБrВr8rrrr9бszHadamardPower._eval_transposecCsFddlm}|jа|б}|jа|б}|а|б}t|||j||ГS)Nr)┌log)Z&sympy.functions.elementary.exponentialrЛrВrHrБZ applyfuncr)r.rJrЛZdexpZlogbaseZdlbaserrrrKеs ■zHadamardPower._eval_derivativecsddlm}ddlm}ddlm}Иjа|б}|D]╚}ddg}ЗfddДt|ГDГ}|j|j }|j|j } t|t|t||gГИjt ИjИjd Гt|| gГgГg|в|jd Н} | jdjdj|_d|_d|_ | jdjdj|_d|_d|_ | g|_q4|S)NrrNrLrP)rrSrTcs$g|]\}}Иjj|dkr|СqSrW)rБr,rXr-rrr╖rz?HadamardPower._eval_derivative_matrix_lines..r)Z validatorrS)rZrOrMr[rQrБr]r\r^r_r`rrВrД┌ _validater'rarbrcrd)r.rJrOrMrQ┌lrrrhrirjrkrr-rr]пs> ¤■ ў Ї z+HadamardPower._eval_derivative_matrix_lines)rlrmrnror%rprБrВr,r/r9rKr]rqrrr)rrА4s8 rАN)Z sympy.corerrZsympy.core.addrZsympy.core.exprrZsympy.core.sortingrZsympy.matrices.commonrr[rZ"sympy.matrices.expressions.specialr r Zsympy.strategiesrrr rrrrrrrCrДrАrrrr┌s n