ó ¡¼™\c@ sÎdZddlmZmZddlmZmZmZmZm Z ddl m Z ddl m Z ddlmZmZmZddlmZddlmZdd lmZmZmZmZmZmZmZmZdd lm Z dd l!m"Z"d d l#m$Z$d dl%m&Z&d dl'm(Z(d„Z)defd„ƒYZ*d„Z+d„Z,d„Z-d„Z.ee.ee,fZ/eed„ee/ŒƒƒZ0d„Z1d„Z2d„Z3d„Z4d„Z5dS(s'Implementation of the Kronecker productiÿÿÿÿ(tdivisiontprint_function(tAddtMultPowtprodtsympify(trange(tadjoint(t MatrixExprt ShapeErrortIdentity(t transpose(t MatrixBase(tcanont conditiont distributetdo_onetexhausttflattenttypedtunpack(t bottom_up(tsifti(tMatAdd(tMatMul(tMatPowcG sM|stdƒ‚nt|Œt|ƒdkr9|dSt|ŒjƒSdS(sf 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.matrices 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.matrices 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 s$Empty Kronecker product is undefinediiN(t TypeErrortvalidatetlentKroneckerProducttdoit(tmatrices((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pytkronecker_products 8 RcB s¡eZdZeZd„Zed„ƒZd„Zd„Z d„Z d„Z d„Z d„Z d „Zd „Zd „Zd „Zd „Zd„Zd„ZRS(sŸ The Kronecker product of two or more arguments. The Kronecker product is a non-commutative product of matrices. Given two matrices of dimension (m, n) and (s, t) it produces a matrix of dimension (m s, n t). This is a symbolic object that simply stores its argument without evaluating it. To actually compute the product, use the function ``kronecker_product()`` or call the the ``.doit()`` or ``.as_explicit()`` methods. >>> from sympy.matrices import KroneckerProduct, MatrixSymbol >>> A = MatrixSymbol('A', 5, 5) >>> B = MatrixSymbol('B', 5, 5) >>> isinstance(KroneckerProduct(A, B), KroneckerProduct) True cO s¬ttt|ƒƒ}td„|Dƒƒrnttd„|Dƒƒƒ}td„|Dƒƒrg|jƒS|Sn|jdtƒ}|r“t |Œnt t |ƒj ||ŒS(Ncs s|]}|jVqdS(N(t is_Identity(t.0ta((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pys mscs s|]}|jVqdS(N(trows(R#R$((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pys nscs s|]}t|tƒVqdS(N(t isinstanceR (R#R$((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pys ostcheck( tlisttmapRtallR Rt as_explicittgettTrueRtsuperRt__new__(tclstargstkwargstretR'((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pyR/ks  cC sR|jdj\}}x/|jdD] }||j9}||j9}q$W||fS(Nii(R1tshapeR%tcols(tselfR%R5tmat((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pyR4ys  cC shd}x[t|jƒD]J}t||jƒ\}}t||jƒ\}}||||f9}qW|S(Ni(treversedR1tdivmodR%R5(R6titjtresultR7tmtn((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pyt_entrys cC s"tttt|jƒƒŒjƒS(N(RR(R)RR1R(R6((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pyt _eval_adjoint‰scC s,tg|jD]}|jƒ^q ŒjƒS(N(RR1t conjugateR(R6R$((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pyt_eval_conjugateŒscC s"tttt|jƒƒŒjƒS(N(RR(R)R R1R(R6((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pyt_eval_transposesc s-ddlm‰t‡fd†|jDƒƒS(Ni(ttracec3 s|]}ˆ|ƒVqdS(N((R#R$(RD(sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pys ”s(RDRR1(R6((RDsClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pyt _eval_trace’sc sbddlm‰m}td„|jDƒƒs9||ƒS|j‰t‡‡fd†|jDƒƒS(Ni(tdett Determinantcs s|]}|jVqdS(N(t is_square(R#R$((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pys ˜sc3 s&|]}ˆ|ƒˆ|jVqdS(N(R%(R#R$(RFR=(sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pys œs(t determinantRFRGR*R1R%R(R6RG((RFR=sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pyt_eval_determinant–s   cC sYy*tg|jD]}|jƒ^qŒSWn(tk rTddlm}||ƒSXdS(Niÿÿÿÿ(tInverse(RR1tinverseR t"sympy.matrices.expressions.inverseRK(R6R$RK((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pyt _eval_inversežs * cC sbt|tƒoa|j|jkoat|jƒt|jƒkoatd„t|j|jƒDƒƒS(s‹Determine whether two matrices have the same Kronecker product structure Examples ======== >>> from sympy import KroneckerProduct, MatrixSymbol, symbols >>> m, n = symbols(r'm, n', integer=True) >>> A = MatrixSymbol('A', m, m) >>> B = MatrixSymbol('B', n, n) >>> C = MatrixSymbol('C', m, m) >>> D = MatrixSymbol('D', n, n) >>> KroneckerProduct(A, B).structurally_equal(KroneckerProduct(C, D)) True >>> KroneckerProduct(A, B).structurally_equal(KroneckerProduct(D, C)) False >>> KroneckerProduct(A, B).structurally_equal(C) False cs s'|]\}}|j|jkVqdS(N(R4(R#R$tb((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pys ¼s(R&RR4RR1R*tzip(R6tother((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pytstructurally_equal¥scC sbt|tƒoa|j|jkoat|jƒt|jƒkoatd„t|j|jƒDƒƒS(sqDetermine 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 cs s'|]\}}|j|jkVqdS(N(R5R%(R#R$RO((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pys Òs(R&RR5R%RR1R*RP(R6RQ((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pythas_matching_shape¾scK s,tttitttƒt6ƒƒ|ƒƒS(N(RRRRRR(R6thints((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pyt_eval_expand_kroneckerproductÔscC sT|j|ƒrH|jgt|j|jƒD]\}}||^q+ŒS||SdS(N(RRt __class__RPR1(R6RQR$RO((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pyt_kronecker_add×s9cC sT|j|ƒrH|jgt|j|jƒD]\}}||^q+ŒS||SdS(N(RSRVRPR1(R6RQR$RO((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pyt_kronecker_mulÝs9cK sY|jdtƒ}|r@g|jD]}|j|^q"}n |j}tt|ŒƒS(Ntdeep(R,R-R1Rt canonicalizeR(R6R2RYtargR1((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pyRãs ( (t__name__t __module__t__doc__R-tis_KroneckerProductR/tpropertyR4R?R@RBRCRERJRNRRRSRURWRXR(((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pyRVs"             cG s)td„|Dƒƒs%tdƒ‚ndS(Ncs s|]}|jVqdS(N(t is_Matrix(R#R[((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pys íss Mix of Matrix and Scalar symbols(R*R(R1((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pyRìscC sg}g}xF|jD];}|jƒ\}}|j|ƒ|jtj|ƒƒqWt|Œ}|dkr{|t|ŒS|S(Ni(R1targs_cnctextendtappendRt _from_argsR(tkrontc_parttnc_partR[tctnc((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pytextract_commutativeós   c G s0td„|Dƒƒs/tdt|ƒƒ‚n|d}x»t|d ƒD]©}|j}|j}xˆt|ƒD]z}||||}x;t|dƒD])}|j|||||dƒ}q˜W|dkrÚ|}qo|j|ƒ}qoW|}qJWt |dd„ƒj } t || ƒr"|S| |ƒSdS( s“Compute 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 cs s|]}t|tƒVqdS(N(R&R (R#R=((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pys .ss&Sequence of Matrices expected, got: %siÿÿÿÿiitkeycS s|jS(N(t_class_priority(tM((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pytJtN( R*RtreprR8R%R5Rtrow_jointcol_jointmaxRVR&( R tmatrix_expansionR7R%R5R:tstartR;tnextt MatrixClass((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pytmatrix_kronecker_products(-   !   cC s*td„|jDƒƒs|St|jŒS(Ncs s|]}t|tƒVqdS(N(R&R (R#R=((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pys Ss(R*R1Ry(Rf((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pytexplicit_kronecker_productQscC s t|tƒS(N(R&R(tx((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pyRo^RpcC s.t|tƒr&td„|jDƒƒSdSdS(Ncs s|]}|jVqdS(N(R4(R#R$((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pys dsi(i(R&RttupleR1(texpr((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pyt_kronecker_dims_keybscC s‹ddlm}t|jtƒ}|jddƒ}|s>|Sg|jƒD]}|d„|ƒ^qK}|syt|ŒSt|Œ|SdS(Niÿÿÿÿ(treduceicS s |j|ƒS(N(RW(R{ty((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pyRopRp(i( t functoolsRRR1R~tpoptNonetvaluesR(R}RR1tnonkronstgrouptkrons((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pytkronecker_mat_addis( cC s©|jƒ\}}d}x€|t|ƒdkrš|||d!\}}t|tƒrt|tƒr|j|ƒ||<|j|dƒq|d7}qW|t|ŒS(Niii(tas_coeff_matricesRR&RRXR‚R(R}tfactorR R:tAtB((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pytkronecker_mat_mulyscC sIt|jtƒrAtg|jjD]}t||jƒ^q"ŒS|SdS(N(R&tbaseRR1Rtexp(R}R$((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pytkronecker_mat_pow‰s/c C s~d„}tttt|titt6tt6tt 6ƒƒƒƒƒ}||ƒ}t |ddƒ}|dk rv|ƒS|SdS(såCombine KronekeckerProduct with expression. If possible write operations on KroneckerProducts of compatible shapes as a single KroneckerProduct. Examples ======== >>> from sympy.matrices.expressions import MatrixSymbol, KroneckerProduct, combine_kronecker >>> from sympy import symbols >>> m, n = symbols(r'm, n', integer=True) >>> A = MatrixSymbol('A', m, n) >>> B = MatrixSymbol('B', n, m) >>> combine_kronecker(KroneckerProduct(A, B)*KroneckerProduct(B, A)) KroneckerProduct(A*B, B*A) >>> combine_kronecker(KroneckerProduct(A, B)+KroneckerProduct(B.T, A.T)) KroneckerProduct(A + B.T, B + A.T) >>> combine_kronecker(KroneckerProduct(A, B)**m) KroneckerProduct(A**m, B**m) cS st|tƒo|jtƒS(N(R&R thasR(R}((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pythaskron¥sRN( RRRRRˆRRRRRtgetattrRƒ(R}R’truleR<R((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pytcombine_kroneckers    N(6R^t __future__RRt sympy.coreRRRRRtsympy.core.compatibilityRtsympy.functionsRt"sympy.matrices.expressions.matexprR R R t$sympy.matrices.expressions.transposeR tsympy.matrices.matricesR tsympy.strategiesRRRRRRRRtsympy.strategies.traverseRtsympy.utilitiesRtmataddRtmatmulRtmatpowRR!RRRkRyRztrulesRZR~RˆRRR•(((sClib/python2.7/site-packages/sympy/matrices/expressions/kronecker.pyts<(: A–   P