\h\4`SrSSKJr SSKJr SSKJrJr SSKJ r SSK J r SSK J r SSKJr SS KJr SS KJr SS KJrJrJrJrJrJrJrJr SS KJr SS KJ r SSK!J"r" SSK#J$r$ SSK%J&r& Sr'"SS\ 5r(Sr)Sr*Sr+Sr,\\,\\*4r-\"\"S\"\-655r.Sr/Sr0Sr1Sr2Sr3g) z'Implementation of the Kronecker product)reduce)prod)Mulsympify)adjoint) ShapeError) MatrixExpr) transpose)Identity) MatrixBase)canon condition distributedo_oneexhaustflattentypedunpack) bottom_up)sift)MatAdd)MatMul)MatPowczU(d [S5e[U5S:XaUS$[U6R5$)a The Kronecker product of two or more arguments. This computes the explicit Kronecker product for subclasses of ``MatrixBase`` i.e. explicit matrices. Otherwise, a symbolic ``KroneckerProduct`` object is returned. Examples ======== For ``MatrixSymbol`` arguments a ``KroneckerProduct`` object is returned. Elements of this matrix can be obtained by indexing, or for MatrixSymbols with known dimension the explicit matrix can be obtained with ``.as_explicit()`` >>> from sympy import kronecker_product, MatrixSymbol >>> A = MatrixSymbol('A', 2, 2) >>> B = MatrixSymbol('B', 2, 2) >>> kronecker_product(A) A >>> kronecker_product(A, B) KroneckerProduct(A, B) >>> kronecker_product(A, B)[0, 1] A[0, 0]*B[0, 1] >>> kronecker_product(A, B).as_explicit() Matrix([ [A[0, 0]*B[0, 0], A[0, 0]*B[0, 1], A[0, 1]*B[0, 0], A[0, 1]*B[0, 1]], [A[0, 0]*B[1, 0], A[0, 0]*B[1, 1], A[0, 1]*B[1, 0], A[0, 1]*B[1, 1]], [A[1, 0]*B[0, 0], A[1, 0]*B[0, 1], A[1, 1]*B[0, 0], A[1, 1]*B[0, 1]], [A[1, 0]*B[1, 0], A[1, 0]*B[1, 1], A[1, 1]*B[1, 0], A[1, 1]*B[1, 1]]]) For explicit matrices the Kronecker product is returned as a Matrix >>> from sympy import Matrix, kronecker_product >>> sigma_x = Matrix([ ... [0, 1], ... [1, 0]]) ... >>> Isigma_y = Matrix([ ... [0, 1], ... [-1, 0]]) ... >>> kronecker_product(sigma_x, Isigma_y) Matrix([ [ 0, 0, 0, 1], [ 0, 0, -1, 0], [ 0, 1, 0, 0], [-1, 0, 0, 0]]) See Also ======== KroneckerProduct z$Empty Kronecker product is undefinedrr) TypeErrorlenKroneckerProductdoit)matricess \/var/www/auris/envauris/lib/python3.13/site-packages/sympy/matrices/expressions/kronecker.pykronecker_productr"s>p >?? 8}{*//11c^\rSrSrSrSrSS.U4Sjjr\S5rSr Sr S r S r S r S rS rSrSrSrSrSrSrSrU=r$)rVaZ The Kronecker product of two or more arguments. The Kronecker product is a non-commutative product of matrices. Given two matrices of dimension (m, n) and (s, t) it produces a matrix of dimension (m s, n t). This is a symbolic object that simply stores its argument without evaluating it. To actually compute the product, use the function ``kronecker_product()`` or call the ``.doit()`` or ``.as_explicit()`` methods. >>> from sympy import KroneckerProduct, MatrixSymbol >>> A = MatrixSymbol('A', 5, 5) >>> B = MatrixSymbol('B', 5, 5) >>> isinstance(KroneckerProduct(A, B), KroneckerProduct) True T)checkc,>[[[U55n[SU55(aD[ [ SU555n[SU55(aUR 5$U$U(a[U6 [TU]$"U/UQ76$)Nc38# UHoRv M g7fN) is_Identity.0as r! +KroneckerProduct.__new__..ms+d}}dc38# UHoRv M g7fr))rowsr+s r!r.r/ns51r0c3B# UHn[U[5v M g7fr) isinstancer r+s r!r.r/os;d:a,,d) listmaprallr r as_explicitvalidatesuper__new__)clsr&argsret __class__s r!r=KroneckerProduct.__new__ks{C&' +d+ + +45556C;d;;;((  dOws*T**r#cURSRupURSSHnXR-nX#R-nM! X4$)Nrr)r?shaper2cols)selfr2rEmats r!rDKroneckerProduct.shapexsMYYq\'' 99QR=C HH D HH D!|r#c Sn[UR5H9n[XR5up[X%R5up'XEXg4-nM; U$Nr)reversedr?divmodr2rE)rFijkwargsresultrGmns r!_entryKroneckerProduct._entrysODII&C!XX&DA!XX&DA !$i F' r#cp[[[[UR556R 5$r))rr7r8rr?rrFs r! _eval_adjointKroneckerProduct._eval_adjoints&c'499&=!>?DDFFr#c[URVs/sHoR5PM sn6R5$s snfr))rr? conjugater)rFr-s r!_eval_conjugate KroneckerProduct._eval_conjugates0!CA++-!CDIIKK!Cs?cp[[[[UR556R 5$r))rr7r8r r?rrVs r!_eval_transpose KroneckerProduct._eval_transposes&c)TYY&?!@AFFHHr#chSSKJn [URVs/sH o!"U5PM sn6$s snf)Nr)trace)rarr?)rFrar-s r! _eval_traceKroneckerProduct._eval_traces* tyy1y!U1Xy1221s/cSSKJnJn [SUR55(dU"U5$UR n[ URVs/sHoA"U5X4R - -PM sn6$s snf)Nr)det Determinantc38# UHoRv M g7fr) is_squarer+s r!r.5KroneckerProduct._eval_determinant..s2 1;; r0) determinantrerfr9r?r2r)rFrerfrQr-s r!_eval_determinant"KroneckerProduct._eval_determinants^12 222t$ $ II;ASVah';<<;s A5c[URVs/sHoR5PM sn6$s snf![a SSKJn U"U5s$f=f)Nr)Inverse)rr?inverser"sympy.matrices.expressions.inversero)rFr-ros r! _eval_inverseKroneckerProduct._eval_inversesH !#499%E9aiik9%EF F%E ! B4=  !s7277AAc4[U[5=(a URUR:H=(ab [UR5[UR5:H=(a0 [ S[ URUR555$)aDetermine whether two matrices have the same Kronecker product structure Examples ======== >>> from sympy import KroneckerProduct, MatrixSymbol, symbols >>> m, n = symbols(r'm, n', integer=True) >>> A = MatrixSymbol('A', m, m) >>> B = MatrixSymbol('B', n, n) >>> C = MatrixSymbol('C', m, m) >>> D = MatrixSymbol('D', n, n) >>> KroneckerProduct(A, B).structurally_equal(KroneckerProduct(C, D)) True >>> KroneckerProduct(A, B).structurally_equal(KroneckerProduct(D, C)) False >>> KroneckerProduct(A, B).structurally_equal(C) False c3X# UH upURUR:Hv M" g7fr)rDr,r-bs r!r.6KroneckerProduct.structurally_equal..s!T9Sv177*9S(*)r5rrDrr?r9ziprFothers r!structurally_equal#KroneckerProduct.structurally_equalsn(5"23UJJ%++-U Nc%**o5UTTYY 9STT Vr#c4[U[5=(a URUR:H=(ab [ UR 5[ UR 5:H=(a0 [ S[UR UR 555$)aDetermine whether two matrices have the appropriate structure to bring matrix multiplication inside the KroneckerProdut Examples ======== >>> from sympy import KroneckerProduct, MatrixSymbol, symbols >>> m, n = symbols(r'm, n', integer=True) >>> A = MatrixSymbol('A', m, n) >>> B = MatrixSymbol('B', n, m) >>> KroneckerProduct(A, B).has_matching_shape(KroneckerProduct(B, A)) True >>> KroneckerProduct(A, B).has_matching_shape(KroneckerProduct(A, B)) False >>> KroneckerProduct(A, B).has_matching_shape(A) False c3X# UH upURUR:Hv M" g7fr))rEr2rws r!r.6KroneckerProduct.has_matching_shape..s!R7QVa!&&(7Qrz)r5rrEr2rr?r9r{r|s r!has_matching_shape#KroneckerProduct.has_matching_shapesn"5"23SII+S Nc%**o5SRs499ejj7QRR Tr#c x[[[[[ [[ 5055"U55$r))rr rrrr)rFhintss r!_eval_expand_kroneckerproduct.KroneckerProduct._eval_expand_kroneckerproducts,uU$4jAQSY6Z#[\]^bcddr#cURU5(aDUR"[URUR5VVs/sH up#X#-PM snn6$X-$s snnfr))r~rAr{r?rFr}r-rxs r!_kronecker_addKroneckerProduct._kronecker_addsT  " "5 ) )>>DIIuzz8R#S8RfqAE8R#ST T< $TA cURU5(aDUR"[URUR5VVs/sH up#X#-PM snn6$X-$s snnfr))rrAr{r?rs r!_kronecker_mulKroneckerProduct._kronecker_mulsT  " "5 ) )>>c$))UZZ6P#Q6PFQAC6P#QR R< $Rrc URSS5nU(a,URVs/sHo3R"S0UD6PM nnO URn[[ U65$s snf)NdeepT)getr?r canonicalizer)rFrrargr?s r!rKroneckerProduct.doitsXyy& 15;#HH%u%D;D99D,d344.s-}}r0z Mix of Matrix and Scalar symbols)r9r)r?s r!r;r;s$ -- - -:;; .r#c/n/nURHKnUR5upEURU5 UR[R "U55 MM [ U6nUS:wa U[ U6-$U$rJ)r?args_cncextendappendr _from_argsr)kronc_partnc_partrcncs r!extract_commutativersp FGyy  as~~b)* &\F {&000 Kr#c [SU55(d[S[U5-5eUSn[USS5HnURnUR n[ U5HZnXXT--n[ US- 5H!nURXXT-U-S--5nM# US:XaUnMIWRU5nM\ WnM [USS9Rn [X5(aU$U "U5$) aCompute the Kronecker product of a sequence of SymPy Matrices. This is the standard Kronecker product of matrices [1]. Parameters ========== matrices : tuple of MatrixBase instances The matrices to take the Kronecker product of. Returns ======= matrix : MatrixBase The Kronecker product matrix. Examples ======== >>> from sympy import Matrix >>> from sympy.matrices.expressions.kronecker import ( ... matrix_kronecker_product) >>> m1 = Matrix([[1,2],[3,4]]) >>> m2 = Matrix([[1,0],[0,1]]) >>> matrix_kronecker_product(m1, m2) Matrix([ [1, 0, 2, 0], [0, 1, 0, 2], [3, 0, 4, 0], [0, 3, 0, 4]]) >>> matrix_kronecker_product(m2, m1) Matrix([ [1, 2, 0, 0], [3, 4, 0, 0], [0, 0, 1, 2], [0, 0, 3, 4]]) References ========== .. [1] https://en.wikipedia.org/wiki/Kronecker_product c3B# UHn[U[5v M g7fr)r4r,rQs r!r.+matrix_kronecker_product..-s;(Qz!Z(((r6z&Sequence of Matrices expected, got: %sNrrcUR$r))_class_priority)Ms r!*matrix_kronecker_product..Is a.?.?r#)key) r9rreprrKr2rErangerow_joincol_joinmaxrAr5) r matrix_expansionrGr2rErMstartrNnext MatrixClasss r!matrix_kronecker_productrs Z ;(; ; ; 4tH~ E   |" &xxxxtA$[0E4!8_$!a%88% Av}}U+ %'(h$?@JJK"00+,,r#cl[SUR55(dU$[UR6$)Nc3B# UHn[U[5v M g7fr)r4rs r!r.-explicit_kronecker_product..Rs<)Qz!Z(()r6)r9r?r)rs r!explicit_kronecker_productrPs+ <$))< < < #TYY //r#c"[U[5$r))r5r)xs r!rr]s :a9I+Jr#cf[U[5(a[SUR55$g)Nc38# UHoRv M g7fr)rvr+s r!r.&_kronecker_dims_key..cs0iWWir0r)r5rtupler?exprs r!_kronecker_dims_keyras($())0dii000r#c[UR[5nURSS5nU(dU$UR 5Vs/sHn[ SU5PM nnU(d[ U6$[ U6U-$s snf)Nrc$URU5$r))r)rys r!r#kronecker_mat_add..ns!1!1!!4r#)rr?rpopvaluesrr)rr?nonkronsgroupkronss r!kronecker_mat_addrhs~  . /Dxxd#H  ++- )'4e <'  ) u~u~((  )sA;cLUR5upSnU[U5S- :atX#US-upE[U[5(a=[U[5(a(UR U5X#'UR US-5 OUS- nU[U5S- :aMtU[ U6-$)Nrr)as_coeff_matricesrr5rrrr)rfactorr rMABs r!kronecker_mat_mulrws--/F A c(ma !A# a) * *z!=M/N/N**1-HK LL1  FA c(ma  &(# ##r#c $[UR[5(ak[SURR55(a@[URRVs/sHn[ XR 5PM sn6$U$s snf)Nc38# UHoRv M g7fr)rhr+s r!r.$kronecker_mat_pow..s6[Nq{{Nr0)r5baserr9r?rexp)rr-s r!kronecker_mat_powrsb$))-..36[DIINN6[3[3[tyy~~!N~!&HH"5~!NOO "Os(B cSn[[[[U[[[ [ [[[055555nU"U5n[USS5nUbU"5$U$)aCombine KronekeckerProduct with expression. If possible write operations on KroneckerProducts of compatible shapes as a single KroneckerProduct. Examples ======== >>> from sympy.matrices.expressions import combine_kronecker >>> from sympy import MatrixSymbol, KroneckerProduct, symbols >>> m, n = symbols(r'm, n', integer=True) >>> A = MatrixSymbol('A', m, n) >>> B = MatrixSymbol('B', n, m) >>> combine_kronecker(KroneckerProduct(A, B)*KroneckerProduct(B, A)) KroneckerProduct(A*B, B*A) >>> combine_kronecker(KroneckerProduct(A, B)+KroneckerProduct(B.T, A.T)) KroneckerProduct(A + B.T, B + A.T) >>> C = MatrixSymbol('C', n, n) >>> D = MatrixSymbol('D', m, m) >>> combine_kronecker(KroneckerProduct(C, D)**m) KroneckerProduct(C**m, D**m) cZ[U[5=(a UR[5$r))r5r hasrrs r!haskron"combine_kronecker..haskrons$ +J9I0JJr#rN) rrrrrrrrrrgetattr)rrrulerPrs r!combine_kroneckerrsu.K ')GU & & & (.)*+ , -D $ZF 664 (D v  r#N)4r functoolsrmathr sympy.corerrsympy.functionsrsympy.matrices.exceptionsr"sympy.matrices.expressions.matexprr $sympy.matrices.expressions.transposer "sympy.matrices.expressions.specialr sympy.matrices.matrixbaser sympy.strategiesr rrrrrrrsympy.strategies.traversersympy.utilitiesrmataddrmatmulrmatpowrr"rr;rrrrulesrrrrrrrr#r!rs-##09:70KKK/ =2@R5zR5j< M-`0  #    y!J!'12  ) $ $r#