ó ~9­\c@ s]dZddlmZmZddlmZddlmZddlm Z ddl m Z ddl m Z mZmZmZmZddlmZdd lmZdd lmZdd lmZdd lmZdd lmZddlmZddl m!Z"ddl#m$Z$ddl%m&Z&de'fd„ƒYZ(de)e(fd„ƒYZ*de*fd„ƒYZ+de,fd„ƒYZ-de-fd„ƒYZ.de-fd„ƒYZ/de-fd„ƒYZ0d e-fd!„ƒYZ1d"e-fd#„ƒYZ2d$e2e1e0e/e.fd%„ƒYZ3d&e,fd'„ƒYZ4d(e,fd)„ƒYZ5d*„Z6e7d+„Z8d,„Z9d-S(.sy Basic methods common to all matrices to be used when creating more advanced matrices (e.g., matrices over rings, etc.). iÿÿÿÿ(tdivisiontprint_function(t defaultdict(t isfunction(trefine(tAtom(tIterabletas_intt is_sequencetrangetreduce(tcall_highest_priority(tExpr(t count_ops(tS(tSymbol(tsympify(tAbs(tsimplify(tSymPyDeprecationWarning(tflattent MatrixErrorcB seZRS((t__name__t __module__(((s4lib/python2.7/site-packages/sympy/matrices/common.pyRst ShapeErrorcB seZdZRS(sWrong matrix shape(RRt__doc__(((s4lib/python2.7/site-packages/sympy/matrices/common.pyR stNonSquareMatrixErrorcB seZRS((RR(((s4lib/python2.7/site-packages/sympy/matrices/common.pyR%stMatrixRequiredcB sPeZdZdZdZdZdZed„ƒZ d„Z d„Z d„Z RS(s_All subclasses of matrix objects must implement the required matrix properties listed here.cO stdƒ‚dS(s†`_new` must, at minimum, be callable as `_new(rows, cols, mat) where mat is a flat list of the elements of the matrix.sSubclasses must implement this.N(tNotImplementedError(tclstargstkwargs((s4lib/python2.7/site-packages/sympy/matrices/common.pyt_new1scC stdƒ‚dS(NsSubclasses must implement this.(R(tselftother((s4lib/python2.7/site-packages/sympy/matrices/common.pyt__eq__8scC stdƒ‚dS(s Implementations of __getitem__ should accept ints, in which case the matrix is indexed as a flat list, tuples (i,j) in which case the (i,j) entry is returned, slices, or mixed tuples (a,b) where a and b are any combintion of slices and integers.sSubclasses must implement this.N(R(R!tkey((s4lib/python2.7/site-packages/sympy/matrices/common.pyt __getitem__;scC stdƒ‚dS(s*The total number of entries in the matrix.sSubclasses must implement this.N(R(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pyt__len__BsN( RRRtNonetrowstcolstshapet _simplifyt classmethodR R#R%R&(((s4lib/python2.7/site-packages/sympy/matrices/common.pyR)s  t MatrixShapingcB s eZdZd„Zd„Zd„Zd„Zd„Zd„Zd„Z d„Z d „Z d „Z d „Z d „Zd „Zd„Zd„Zd„Zed„ƒZd„Zd„Zd„Zd„Zd„Zed„ƒZd„Zd„Zed„ƒZRS(s;Provides basic matrix shaping and extracting of submatricesc s/‡‡fd†}ˆjˆjˆjd|ƒS(Nc s,|ˆkrˆ||fSˆ||dfS(Ni((titj(tcolR!(s4lib/python2.7/site-packages/sympy/matrices/common.pytentryKsi(R R(R)(R!R0R1((R0R!s4lib/python2.7/site-packages/sympy/matrices/common.pyt _eval_col_delJsc sGˆj}‡‡‡fd†‰ˆjˆjˆjˆj‡fd†ƒS(Nc sd|ˆkrˆ||fSˆ|ko8ˆˆjknrOˆ||ˆfSˆ||ˆjfS(N(R)(R.R/(R"tposR!(s4lib/python2.7/site-packages/sympy/matrices/common.pyR1Rs  #c s ˆ||ƒS(N((R.R/(R1(s4lib/python2.7/site-packages/sympy/matrices/common.pytZt(R)R R((R!R3R"R)((R1R"R3R!s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_col_insertOs c sPˆj‰‡‡‡fd†‰tˆˆƒjˆjˆjˆj‡fd†ƒS(Nc s,|ˆkrˆ||fSˆ|ˆ|fS(N((R.R/(R"R(R!(s4lib/python2.7/site-packages/sympy/matrices/common.pyR1_s c s ˆ||ƒS(N((R.R/(R1(s4lib/python2.7/site-packages/sympy/matrices/common.pyR4eR5(R(tclassofR R)(R!R"((R1R"R(R!s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_col_join\s "c sct|ƒ‰|j‰‡‡fd†|Dƒ}|jt|ƒtˆƒt‡fd†|DƒƒƒS(Nc3 s*|] }ˆD]}|ˆ|Vq qdS(N((t.0R.R/(R)tcolsList(s4lib/python2.7/site-packages/sympy/matrices/common.pys jsc3 s|]}ˆ|VqdS(N((R9R.(tmat(s4lib/python2.7/site-packages/sympy/matrices/common.pys ls(tlistR)R tlen(R!trowsListR:tindices((R)R:R;s4lib/python2.7/site-packages/sympy/matrices/common.pyt _eval_extractgs   c s&g‰‡‡fd†‰ˆ|ƒˆS(Nc s2d}x%||jdkr-|dkrW|d|d…f}||d…df}n8|d|…|d…f}||d…d|…f}t|ƒs§t|ƒr·|d7}q q ˆj|d|…d|…fƒ|j|d|…d|…fjkrdSˆ||d…|d…fƒdSq WdS(Nii(R*tanytappend(tMR.t to_the_rightt to_the_bottom(trecurse_sub_blockst sub_blocks(s4lib/python2.7/site-packages/sympy/matrices/common.pyRFqs  #( ((R!((RFRGs4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_get_diag_blocksns c s/‡‡fd†}ˆjˆjdˆj|ƒS(Nc s,|ˆkrˆ||fSˆ|d|fS(Ni((R.R/(trowR!(s4lib/python2.7/site-packages/sympy/matrices/common.pyR1‰si(R R(R)(R!RIR1((RIR!s4lib/python2.7/site-packages/sympy/matrices/common.pyt _eval_row_delˆscC sLt|ƒ}||j}t|ƒ|||+|j|j|j|j|ƒS(N(R<R)R R((R!R3R"tentriest insert_pos((s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_row_inserts  c sPˆj‰‡‡‡fd†‰tˆˆƒjˆjˆjˆj‡fd†ƒS(Nc s,|ˆkrˆ||fSˆ||ˆfS(N((R.R/(R)R"R!(s4lib/python2.7/site-packages/sympy/matrices/common.pyR1–s c s ˆ||ƒS(N((R.R/(R1(s4lib/python2.7/site-packages/sympy/matrices/common.pyR4œR5(R)R7R R((R!R"((R)R1R"R!s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_row_join“s "cC s6gt|jƒD]"}t||dd…fƒ^qS(N(R R(R<(R!R.((s4lib/python2.7/site-packages/sympy/matrices/common.pyt _eval_tolistžsc s4ˆj‰‡‡fd†}ˆjtˆƒd|ƒS(Nc s&|ˆ}||ˆ}ˆ||fS(N((tnt_R/R.(R(R!(s4lib/python2.7/site-packages/sympy/matrices/common.pyR1¤s i(R(R R=(R!R1((R(R!s4lib/python2.7/site-packages/sympy/matrices/common.pyt _eval_vec¡s cC s`|dkr||j7}nd|ko6|jknsStdj|ƒƒ‚n|j|ƒS(sDelete the specified column.isColumn {} out of range.(R)t ValueErrortformatR2(R!R0((s4lib/python2.7/site-packages/sympy/matrices/common.pytcol_del¬s  cC sŸ|st|ƒ|ƒSt|ƒ}|dkr>|j|}n|dkrSd}n||jkrn|j}n|j|jkrtdƒ‚n|j||ƒS(syInsert one or more columns at the given column position. Examples ======== >>> from sympy import zeros, ones >>> M = zeros(3) >>> V = ones(3, 1) >>> M.col_insert(1, V) Matrix([ [0, 1, 0, 0], [0, 1, 0, 0], [0, 1, 0, 0]]) See Also ======== col row_insert is5`self` and `other` must have the same number of rows.(ttypeRR)R(RR6(R!R3R"((s4lib/python2.7/site-packages/sympy/matrices/common.pyt col_insert´s      cC sn|jdkr@|j|jkr@|jd|jgƒj|ƒS|j|jkratdƒ‚n|j|ƒS(s†Concatenates two matrices along self's last and other's first row. Examples ======== >>> from sympy import zeros, ones >>> M = zeros(3) >>> V = ones(1, 3) >>> M.col_join(V) Matrix([ [0, 0, 0], [0, 0, 0], [0, 0, 0], [1, 1, 1]]) See Also ======== col row_join is8`self` and `other` must have the same number of columns.(R(R)R tcol_joinRR8(R!R"((s4lib/python2.7/site-packages/sympy/matrices/common.pyRXÜs ! cC s|dd…|fS(s4Elementary column selector. Examples ======== >>> from sympy import eye >>> eye(2).col(0) Matrix([ [1], [0]]) See Also ======== row col_op col_swap col_del col_join col_insert N((R!R/((s4lib/python2.7/site-packages/sympy/matrices/common.pyR0ûscC st|ƒ st|ƒ r)tdƒ‚n|rstd„|Dƒƒrsgt|ƒD]\}}|rR|^qR}n|r½td„|Dƒƒr½gt|ƒD]\}}|rœ|^qœ}ng|D]}t||jƒ^qÄ}g|D]}t||jƒ^qé}|j||ƒS(sReturn a submatrix by specifying a list of rows and columns. Negative indices can be given. All indices must be in the range -n <= i < n where n is the number of rows or columns. Examples ======== >>> from sympy import Matrix >>> m = Matrix(4, 3, range(12)) >>> m Matrix([ [0, 1, 2], [3, 4, 5], [6, 7, 8], [9, 10, 11]]) >>> m.extract([0, 1, 3], [0, 1]) Matrix([ [0, 1], [3, 4], [9, 10]]) Rows or columns can be repeated: >>> m.extract([0, 0, 1], [-1]) Matrix([ [2], [2], [5]]) Every other row can be taken by using range to provide the indices: >>> m.extract(range(0, m.rows, 2), [-1]) Matrix([ [2], [8]]) RowsList or colsList can also be a list of booleans, in which case the rows or columns corresponding to the True values will be selected: >>> m.extract([0, 1, 2, 3], [True, False, True]) Matrix([ [0, 2], [3, 5], [6, 8], [9, 11]]) s&rowsList and colsList must be iterablecs s|]}t|tƒVqdS(N(t isinstancetbool(R9R.((s4lib/python2.7/site-packages/sympy/matrices/common.pys Fscs s|]}t|tƒVqdS(N(RYRZ(R9R.((s4lib/python2.7/site-packages/sympy/matrices/common.pys Hs(Rt TypeErrortallt enumerateta2idxR(R)R@(R!R>R:tindextitemtk((s4lib/python2.7/site-packages/sympy/matrices/common.pytextracts0..%%cC s |jƒS(szObtains the square sub-matrices on the main diagonal of a square matrix. Useful for inverting symbolic matrices or solving systems of linear equations which may be decoupled by having a block diagonal structure. Examples ======== >>> from sympy import Matrix >>> from sympy.abc import x, y, z >>> A = Matrix([[1, 3, 0, 0], [y, z*z, 0, 0], [0, 0, x, 0], [0, 0, 0, 0]]) >>> a1, a2, a3 = A.get_diag_blocks() >>> a1 Matrix([ [1, 3], [y, z**2]]) >>> a2 Matrix([[x]]) >>> a3 Matrix([[0]]) (RH(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pytget_diag_blocksQscG s<t|ƒdkr|jƒSt|dƒ}t|j|ƒS(s1Return a matrix formed by joining args horizontally (i.e. by repeated application of row_join). Examples ======== >>> from sympy.matrices import Matrix, eye >>> Matrix.hstack(eye(2), 2*eye(2)) Matrix([ [1, 0, 2, 0], [0, 1, 0, 2]]) i(R=R RVR trow_join(RRtkls((s4lib/python2.7/site-packages/sympy/matrices/common.pythstackks c sRˆjˆj|ˆkr3td|ˆfƒ‚nˆj|ˆ‡‡fd†ƒS(s§Reshape the matrix. Total number of elements must remain the same. Examples ======== >>> from sympy import Matrix >>> m = Matrix(2, 3, lambda i, j: 1) >>> m Matrix([ [1, 1, 1], [1, 1, 1]]) >>> m.reshape(1, 6) Matrix([[1, 1, 1, 1, 1, 1]]) >>> m.reshape(3, 2) Matrix([ [1, 1], [1, 1], [1, 1]]) s Invalid reshape parameters %d %dc sˆ|ˆ|S(N((R.R/(R)R!(s4lib/python2.7/site-packages/sympy/matrices/common.pyR4–R5(R(R)RSR (R!R(R)((R)R!s4lib/python2.7/site-packages/sympy/matrices/common.pytreshapescC s`|dkr||j7}nd|ko6|jknsStdj|ƒƒ‚n|j|ƒS(sDelete the specified row.isRow {} out of range.(R(RSRTRJ(R!RI((s4lib/python2.7/site-packages/sympy/matrices/common.pytrow_del˜s  cC sœ|s|j|ƒSt|ƒ}|dkr;|j|}n|dkrPd}n||jkrk|j}n|j|jkrŒtdƒ‚n|j||ƒS(s}Insert one or more rows at the given row position. Examples ======== >>> from sympy import zeros, ones >>> M = zeros(3) >>> V = ones(1, 3) >>> M.row_insert(1, V) Matrix([ [0, 0, 0], [1, 1, 1], [0, 0, 0], [0, 0, 0]]) See Also ======== row col_insert is8`self` and `other` must have the same number of columns.(R RR(R)RRM(R!R3R"((s4lib/python2.7/site-packages/sympy/matrices/common.pyt row_insert¡s       cC sn|jdkr@|j|jkr@|j|jdgƒj|ƒS|j|jkratdƒ‚n|j|ƒS(s|Concatenates two matrices along self's last and rhs's first column Examples ======== >>> from sympy import zeros, ones >>> M = zeros(3) >>> V = ones(3, 1) >>> M.row_join(V) Matrix([ [0, 0, 0, 1], [0, 0, 0, 1], [0, 0, 0, 1]]) See Also ======== row col_join is3`self` and `rhs` must have the same number of rows.(R)R(R RdRRN(R!R"((s4lib/python2.7/site-packages/sympy/matrices/common.pyRdÊs ! cC s||dd…fS(sElementary row selector. Examples ======== >>> from sympy import eye >>> eye(2).row(0) Matrix([[1, 0]]) See Also ======== col row_op row_swap row_del row_join row_insert N((R!R.((s4lib/python2.7/site-packages/sympy/matrices/common.pyRIèscC s|j|jfS(sThe shape (dimensions) of the matrix as the 2-tuple (rows, cols). Examples ======== >>> from sympy.matrices import zeros >>> M = zeros(2, 3) >>> M.shape (2, 3) >>> M.rows 2 >>> M.cols 3 (R(R)(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pyR*þscC s@|js gS|js6gt|jƒD] }g^q&S|jƒS(s Return the Matrix as a nested Python list. Examples ======== >>> from sympy import Matrix, ones >>> m = Matrix(3, 3, range(9)) >>> m Matrix([ [0, 1, 2], [3, 4, 5], [6, 7, 8]]) >>> m.tolist() [[0, 1, 2], [3, 4, 5], [6, 7, 8]] >>> ones(3, 0).tolist() [[], [], []] When there are no rows then it will not be possible to tell how many columns were in the original matrix: >>> ones(0, 3).tolist() [] (R(R)R RO(R!R.((s4lib/python2.7/site-packages/sympy/matrices/common.pyttolists    cC s |jƒS(sŒReturn the Matrix converted into a one column matrix by stacking columns Examples ======== >>> from sympy import Matrix >>> m=Matrix([[1, 3], [2, 4]]) >>> m Matrix([ [1, 3], [2, 4]]) >>> m.vec() Matrix([ [1], [2], [3], [4]]) See Also ======== vech (RR(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pytvec/scG s<t|ƒdkr|jƒSt|dƒ}t|j|ƒS(sCReturn a matrix formed by joining args vertically (i.e. by repeated application of col_join). Examples ======== >>> from sympy.matrices import Matrix, eye >>> Matrix.vstack(eye(2), 2*eye(2)) Matrix([ [1, 0], [0, 1], [2, 0], [0, 2]]) i(R=R RVR RX(RRRe((s4lib/python2.7/site-packages/sympy/matrices/common.pytvstackIs (RRRR2R6R8R@RHRJRMRNRORRRURWRXR0RbRcR,RfRgRhRiRdRItpropertyR*RjRkRl(((s4lib/python2.7/site-packages/sympy/matrices/common.pyR-Gs6        (   >   )    t MatrixSpecialcB s¶eZdZed„ƒZed„ƒZedd„ƒZed„ƒZed„ƒZed„ƒZ ed d„ƒZ ed d d „ƒZ ed d „ƒZ ed d „ƒZRS( s Construction of special matricesc s"‡fd†}|j|||ƒS(sUdiag_dict is a defaultdict containing all the entries of the diagonal matrix.c sˆ||fS(N((R.R/(t diag_dict(s4lib/python2.7/site-packages/sympy/matrices/common.pyR1gs(R (RR(R)RoR1((Ros4lib/python2.7/site-packages/sympy/matrices/common.pyt _eval_diagcscC sd„}|j|||ƒS(NcS s||krtjStjS(N(RtOnetZero(R.R/((s4lib/python2.7/site-packages/sympy/matrices/common.pyR1ms(R (RR(R)R1((s4lib/python2.7/site-packages/sympy/matrices/common.pyt _eval_eyeks tupperc s@|dkr‡fd†}n‡fd†}|j|||ƒS(Ntlowerc s.||krˆS|d|kr'tjStjS(Ni(RRqRr(R.R/(t eigenvalue(s4lib/python2.7/site-packages/sympy/matrices/common.pyR1ts  c s.||krˆS|d|kr'tjStjS(Ni(RRqRr(R.R/(Rv(s4lib/python2.7/site-packages/sympy/matrices/common.pyR1{s  (R (RR(R)RvtbandR1((Rvs4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_jordan_blockqs cC sd„}|j|||ƒS(NcS stjS(N(RRq(R.R/((s4lib/python2.7/site-packages/sympy/matrices/common.pyR1…s(R (RR(R)R1((s4lib/python2.7/site-packages/sympy/matrices/common.pyt _eval_onesƒs cC sd„}|j|||ƒS(NcS stjS(N(RRr(R.R/((s4lib/python2.7/site-packages/sympy/matrices/common.pyR1‹s(R (RR(R)R1((s4lib/python2.7/site-packages/sympy/matrices/common.pyt _eval_zeros‰s c sÛ|jd|ƒ}t|ƒdkrXt|dƒrXt|ddtƒ rX|d}nd„‰t‡fd†|Dƒƒ}t‡fd†|Dƒƒ}|jd|ƒ}|jd |ƒ}||ksÕ||krötd j||||ƒƒ‚ntd „ƒ}d \} } x´|D]¬} t | dƒr xSt | j ƒD]B} x9t | j ƒD](} | | | f|| | | | f>> from sympy.matrices import Matrix >>> Matrix.diag(1, 2, 3) Matrix([ [1, 0, 0], [0, 2, 0], [0, 0, 3]]) >>> Matrix.diag([1, 2, 3]) Matrix([ [1, 0, 0], [0, 2, 0], [0, 0, 3]]) The diagonal elements can be matrices; diagonal filling will continue on the diagonal from the last element of the matrix: >>> from sympy.abc import x, y, z >>> a = Matrix([x, y, z]) >>> b = Matrix([[1, 2], [3, 4]]) >>> c = Matrix([[5, 6]]) >>> Matrix.diag(a, 7, b, c) Matrix([ [x, 0, 0, 0, 0, 0], [y, 0, 0, 0, 0, 0], [z, 0, 0, 0, 0, 0], [0, 7, 0, 0, 0, 0], [0, 0, 1, 2, 0, 0], [0, 0, 3, 4, 0, 0], [0, 0, 0, 0, 5, 6]]) A given band off the diagonal can be made by padding with a vertical or horizontal "kerning" vector: >>> hpad = Matrix(0, 2, []) >>> vpad = Matrix(2, 0, []) >>> Matrix.diag(vpad, 1, 2, 3, hpad) + Matrix.diag(hpad, 4, 5, 6, vpad) Matrix([ [0, 0, 4, 0, 0], [0, 0, 0, 5, 0], [1, 0, 0, 0, 6], [0, 2, 0, 0, 0], [0, 0, 3, 0, 0]]) The type of the resulting matrix can be affected with the ``cls`` keyword. >>> type(Matrix.diag(1)) >>> from sympy.matrices import ImmutableMatrix >>> type(Matrix.diag(1, cls=ImmutableMatrix)) Riit is_MatrixcS s#t|dƒr|j|jfSdS(s&Compute the size of the diagonal blockR(i(ii(thasattrR(R)(tm((s4lib/python2.7/site-packages/sympy/matrices/common.pytsizeÜsc3 s|]}ˆ|ƒdVqdS(iN((R9R}(R~(s4lib/python2.7/site-packages/sympy/matrices/common.pys ásc3 s|]}ˆ|ƒdVqdS(iN((R9R}(R~(s4lib/python2.7/site-packages/sympy/matrices/common.pys âsR(R)sOA {} x {} diagnal matrix cannot accommodate adiagonal of size at least {} x {}.cS stjS(N(RRr(((s4lib/python2.7/site-packages/sympy/matrices/common.pyR4ëR5(ii(tgetR=RtgetattrtFalsetsumRSRTRR|R R(R)Rp(ReRRtklasst diag_rowst diag_colsR(R)t diag_entriestrow_postcol_posR}R.R/((R~s4lib/python2.7/site-packages/sympy/matrices/common.pytdiags2H9     *  cK sP|dkr|}n|jd|ƒ}t|ƒt|ƒ}}|j||ƒS(sãReturns an identity matrix. Args ==== rows : rows of the matrix cols : cols of the matrix (if None, cols=rows) kwargs ====== cls : class of the returned matrix RN(R'RRRs(ReR(R)RRƒ((s4lib/python2.7/site-packages/sympy/matrices/common.pyteyeüs   c K sÀd|ksd|kr@tddddddd d ƒjƒn|jd |ƒ}|jd d ƒ}|jddƒ}|jddƒ}|jddƒ}|dkrÁ|dkrÁtdƒ‚nN||krúd||fkrútdj||ƒƒ‚n|dk r|}n|||fdkr3tdƒ‚n|dk rO||}}nB|dk rp|dkrp|}n!|dk r‘|dkr‘|}nt|ƒt|ƒ}}|j||||ƒS(s Returns a Jordan block Parameters ========== size : Integer, optional Specifies the shape of the Jordan block matrix. eigenvalue : Number or Symbol Specifies the value for the main diagonal of the matrix. .. note:: The keyword ``eigenval`` is also specified as an alias of this keyword, but it is not recommended to use. We may deprecate the alias in later release. band : 'upper' or 'lower', optional Specifies the position of the off-diagonal to put `1` s on. cls : Matrix, optional Specifies the matrix class of the output form. If it is not specified, the class type where the method is being executed on will be returned. rows, cols : Integer, optional Specifies the shape of the Jordan block matrix. See Notes section for the details of how these key works. .. note:: This feature will be deprecated in the future. Returns ======= Matrix A Jordan block matrix. Raises ====== ValueError If insufficient arguments are given for matrix size specification, or no eigenvalue is given. Examples ======== Creating a default Jordan block: >>> from sympy import Matrix >>> from sympy.abc import x >>> Matrix.jordan_block(4, x) Matrix([ [x, 1, 0, 0], [0, x, 1, 0], [0, 0, x, 1], [0, 0, 0, x]]) Creating an alternative Jordan block matrix where `1` is on lower off-diagonal: >>> Matrix.jordan_block(4, x, band='lower') Matrix([ [x, 0, 0, 0], [1, x, 0, 0], [0, 1, x, 0], [0, 0, 1, x]]) Creating a Jordan block with keyword arguments >>> Matrix.jordan_block(size=4, eigenvalue=x) Matrix([ [x, 1, 0, 0], [0, x, 1, 0], [0, 0, x, 1], [0, 0, 0, x]]) Notes ===== .. note:: This feature will be deprecated in the future. The keyword arguments ``size``, ``rows``, ``cols`` relates to the Jordan block size specifications. If you want to create a square Jordan block, specify either one of the three arguments. If you want to create a rectangular Jordan block, specify ``rows`` and ``cols`` individually. +--------------------------------+---------------------+ | Arguments Given | Matrix Shape | +----------+----------+----------+----------+----------+ | size | rows | cols | rows | cols | +==========+==========+==========+==========+==========+ | size | Any | size | size | +----------+----------+----------+----------+----------+ | | None | ValueError | | +----------+----------+----------+----------+ | None | rows | None | rows | rows | | +----------+----------+----------+----------+ | | None | cols | cols | cols | + +----------+----------+----------+----------+ | | rows | cols | rows | cols | +----------+----------+----------+----------+----------+ References ========== .. [1] https://en.wikipedia.org/wiki/Jordan_matrix R(R)tfeatures"Keyword arguments 'rows' or 'cols'tissueiæ>t useinsteads(a more generic banded matrix constructortdeprecated_since_versions1.4RRwRtteigenvalsMust supply an eigenvalues=Inconsistent values are given: 'eigenval'={}, 'eigenvalue'={}sMust supply a matrix sizeN(NNN( RtwarntpopR'RRSRTRRx( ReR~RvRRƒRwR(R)R((s4lib/python2.7/site-packages/sympy/matrices/common.pyt jordan_blocks:v     cK sP|dkr|}n|jd|ƒ}t|ƒt|ƒ}}|j||ƒS(sáReturns a matrix of ones. Args ==== rows : rows of the matrix cols : cols of the matrix (if None, cols=rows) kwargs ====== cls : class of the returned matrix RN(R'RRRy(ReR(R)RRƒ((s4lib/python2.7/site-packages/sympy/matrices/common.pytones­s   cK sP|dkr|}n|jd|ƒ}t|ƒt|ƒ}}|j||ƒS(sâReturns a matrix of zeros. Args ==== rows : rows of the matrix cols : cols of the matrix (if None, cols=rows) kwargs ====== cls : class of the returned matrix RN(R'RRRz(ReR(R)RRƒ((s4lib/python2.7/site-packages/sympy/matrices/common.pytzerosÂs   N(RRRR,RpRsRxRyRzR‰R'RŠR’R“R”(((s4lib/python2.7/site-packages/sympy/matrices/common.pyRn`s m›tMatrixPropertiescB s[eZdZd„Zd„Zd„Zd„Zd„Zd„Zd„Z d„Z d „Z d „Z d „Z d „Zd „Zd„Zd„Zed„ƒZd„Zed„Zd„Zeed„ƒZed„ƒZed„ƒZed„ƒZed„ƒZd„Zed„Zed„ƒZed„ƒZ ed„ƒZ!d„Z"RS(s&Provides basic properties of a matrix.cG s4tƒ}x$|D]}|j|j|ŒƒqW|S(N(tsettupdatetatoms(R!ttypestresultR.((s4lib/python2.7/site-packages/sympy/matrices/common.pyt _eval_atomsÛs  cC stƒjd„|DƒŒS(Ncs s|]}|jVqdS(N(t free_symbols(R9R.((s4lib/python2.7/site-packages/sympy/matrices/common.pys âs(R–tunion(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_free_symbolsásc st‡fd†|DƒƒS(Nc3 s|]}|jˆŒVqdS(N(thas(R9ta(tpatterns(s4lib/python2.7/site-packages/sympy/matrices/common.pys ås(RA(R!R¡((R¡s4lib/python2.7/site-packages/sympy/matrices/common.pyt _eval_hasäsc s0t‡‡fd†tˆjƒDƒƒs,tStS(Nc3 sL|]B}tˆjƒD],}ˆˆ||fˆ||fƒjVqqdS(N(R R)tis_zero(R9R.R/(R!tsimpfunc(s4lib/python2.7/site-packages/sympy/matrices/common.pys ès(R\R R(RtTrue(R!R¤((R!R¤s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_is_anti_symmetricçs(cC sXxQt|jƒD]@}x7t|jƒD]&}||kr&|||fr&tSq&WqWtS(N(R R(R)RR¥(R!R.R/((s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_is_diagonalìs  c s.ˆjˆjˆj‡‡fd†ƒ}|jS(Nc s(ˆˆ||fˆ||fjƒƒS(N(t conjugate(R.R/(R!R¤(s4lib/python2.7/site-packages/sympy/matrices/common.pyR4÷R5(R R(R)R£(R!R¤R;((R!R¤s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_is_matrix_hermitianös'c s/d„‰t‡‡fd†tˆjƒDƒƒS(NcS s||krdSdS(Nii((R.R/((s4lib/python2.7/site-packages/sympy/matrices/common.pytdiracûs c3 sD|]:}tˆjƒD]$}ˆ||fˆ||ƒkVqqdS(N(R R)(R9R.R/(RªR!(s4lib/python2.7/site-packages/sympy/matrices/common.pys s(R\R R((R!((RªR!s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_is_Identityús c s#t‡fd†tˆjƒDƒƒS(Nc3 s?|]5}t|dˆjƒD]}ˆ||fjVqqdS(iN(R R)R£(R9R.R/(R!(s4lib/python2.7/site-packages/sympy/matrices/common.pys s(R\R R((R!((R!s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_is_lower_hessenbergsc s#t‡fd†tˆjƒDƒƒS(Nc3 s?|]5}t|dˆjƒD]}ˆ||fjVqqdS(iN(R R)R£(R9R.R/(R!(s4lib/python2.7/site-packages/sympy/matrices/common.pys s(R\R R((R!((R!s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_is_lowerscC s |jtƒS(N(RŸR(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_is_symbolic sc s.ˆjˆjˆj‡‡fd†ƒ}|jS(Nc s"ˆˆ||fˆ||fƒS(N((R.R/(R!R¤(s4lib/python2.7/site-packages/sympy/matrices/common.pyR4R5(R R(R)R£(R!R¤R;((R!R¤s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_is_symmetrics'cC s8td„|DƒƒrtStd„|Dƒƒr4dStS(Ncs s|]}|jtkVqdS(N(R£R(R9R.((s4lib/python2.7/site-packages/sympy/matrices/common.pys scs s|]}|jdkVqdS(N(R£R'(R9R.((s4lib/python2.7/site-packages/sympy/matrices/common.pys s(RARR'R¥(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pyt _eval_is_zeros c s&t‡fd†tdˆjƒDƒƒS(Nc3 sE|];}ttˆj|dƒƒD]}ˆ||fjVq#qdS(iN(R tminR)R£(R9R.R/(R!(s4lib/python2.7/site-packages/sympy/matrices/common.pys si(R\R R((R!((R!s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_is_upper_hessenbergscC s g|D]}|js|^qS(N(R£(R!R.((s4lib/python2.7/site-packages/sympy/matrices/common.pyt _eval_values scG s5td„|Dƒƒ}|s(tf}n|j|ŒS(sReturns the atoms that form the current object. Examples ======== >>> from sympy.abc import x, y >>> from sympy.matrices import Matrix >>> Matrix([[x]]) Matrix([[x]]) >>> _.atoms() {x} cs s0|]&}t|tƒr|n t|ƒVqdS(N(RYRV(R9tt((s4lib/python2.7/site-packages/sympy/matrices/common.pys 1s(ttupleRR›(R!R™((s4lib/python2.7/site-packages/sympy/matrices/common.pyR˜#s cC s |jƒS(sâReturns the free symbols within the matrix. Examples ======== >>> from sympy.abc import x >>> from sympy.matrices import Matrix >>> Matrix([[x], [1]]).free_symbols {x} (Rž(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pyRœ6s cG s |j|ŒS(sûTest whether any subexpression matches any of the patterns. Examples ======== >>> from sympy import Matrix, SparseMatrix, Float >>> from sympy.abc import x, y >>> A = Matrix(((1, x), (0.2, 3))) >>> B = SparseMatrix(((1, x), (0.2, 3))) >>> A.has(x) True >>> A.has(y) False >>> A.has(Float) True >>> B.has(x) True >>> B.has(y) False >>> B.has(Float) True (R¢(R!R¡((s4lib/python2.7/site-packages/sympy/matrices/common.pyRŸDscC sD|}t|ƒs*|rtnd„}n|js7tS|j|ƒS(sCheck if matrix M is an antisymmetric matrix, that is, M is a square matrix with all M[i, j] == -M[j, i]. When ``simplify=True`` (default), the sum M[i, j] + M[j, i] is simplified before testing to see if it is zero. By default, the SymPy simplify function is used. To use a custom function set simplify to a function that accepts a single argument which returns a simplified expression. To skip simplification, set simplify to False but note that although this will be faster, it may induce false negatives. Examples ======== >>> from sympy import Matrix, symbols >>> m = Matrix(2, 2, [0, 1, -1, 0]) >>> m Matrix([ [ 0, 1], [-1, 0]]) >>> m.is_anti_symmetric() True >>> x, y = symbols('x y') >>> m = Matrix(2, 3, [0, 0, x, -y, 0, 0]) >>> m Matrix([ [ 0, 0, x], [-y, 0, 0]]) >>> m.is_anti_symmetric() False >>> from sympy.abc import x, y >>> m = Matrix(3, 3, [0, x**2 + 2*x + 1, y, ... -(x + 1)**2 , 0, x*y, ... -y, -x*y, 0]) Simplification of matrix elements is done by default so even though two elements which should be equal and opposite wouldn't pass an equality test, the matrix is still reported as anti-symmetric: >>> m[0, 1] == -m[1, 0] False >>> m.is_anti_symmetric() True If 'simplify=False' is used for the case when a Matrix is already simplified, this will speed things up. Here, we see that without simplification the matrix does not appear anti-symmetric: >>> m.is_anti_symmetric(simplify=False) False But if the matrix were already expanded, then it would appear anti-symmetric and simplification in the is_anti_symmetric routine is not needed: >>> m = m.expand() >>> m.is_anti_symmetric(simplify=False) True cS s|S(N((tx((s4lib/python2.7/site-packages/sympy/matrices/common.pyR4žR5(RR+t is_squareRR¦(R!RR¤((s4lib/python2.7/site-packages/sympy/matrices/common.pytis_anti_symmetric]s ?  cC s |jƒS(sCheck if matrix is diagonal, that is matrix in which the entries outside the main diagonal are all zero. Examples ======== >>> from sympy import Matrix, diag >>> m = Matrix(2, 2, [1, 0, 0, 2]) >>> m Matrix([ [1, 0], [0, 2]]) >>> m.is_diagonal() True >>> m = Matrix(2, 2, [1, 1, 0, 2]) >>> m Matrix([ [1, 1], [0, 2]]) >>> m.is_diagonal() False >>> m = diag(1, 2, 3) >>> m Matrix([ [1, 0, 0], [0, 2, 0], [0, 0, 3]]) >>> m.is_diagonal() True See Also ======== is_lower is_upper is_diagonalizable diagonalize (R§(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pyt is_diagonal¤s)cC sD|js tS|}t|ƒs7|r+tnd„}n|j|ƒS(spChecks if the matrix is Hermitian. In a Hermitian matrix element i,j is the complex conjugate of element j,i. Examples ======== >>> from sympy.matrices import Matrix >>> from sympy import I >>> from sympy.abc import x >>> a = Matrix([[1, I], [-I, 1]]) >>> a Matrix([ [ 1, I], [-I, 1]]) >>> a.is_hermitian True >>> a[0, 0] = 2*I >>> a.is_hermitian False >>> a[0, 0] = x >>> a.is_hermitian >>> a[0, 1] = a[1, 0]*I >>> a.is_hermitian False cS s|S(N((R¶((s4lib/python2.7/site-packages/sympy/matrices/common.pyR4ñR5(R·RRR+R©(R!RR¤((s4lib/python2.7/site-packages/sympy/matrices/common.pyt is_hermitianÏs   cC s|js tS|jƒS(N(R·RR«(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pyt is_Identityõs cC s |jƒS(s6Checks if the matrix is in the lower-Hessenberg form. The lower hessenberg matrix has zero entries above the first superdiagonal. Examples ======== >>> from sympy.matrices import Matrix >>> a = Matrix([[1, 2, 0, 0], [5, 2, 3, 0], [3, 4, 3, 7], [5, 6, 1, 1]]) >>> a Matrix([ [1, 2, 0, 0], [5, 2, 3, 0], [3, 4, 3, 7], [5, 6, 1, 1]]) >>> a.is_lower_hessenberg True See Also ======== is_upper_hessenberg is_lower (R¬(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pytis_lower_hessenbergûscC s |jƒS(smCheck if matrix is a lower triangular matrix. True can be returned even if the matrix is not square. Examples ======== >>> from sympy import Matrix >>> m = Matrix(2, 2, [1, 0, 0, 1]) >>> m Matrix([ [1, 0], [0, 1]]) >>> m.is_lower True >>> m = Matrix(4, 3, [0, 0, 0, 2, 0, 0, 1, 4 , 0, 6, 6, 5]) >>> m Matrix([ [0, 0, 0], [2, 0, 0], [1, 4, 0], [6, 6, 5]]) >>> m.is_lower True >>> from sympy.abc import x, y >>> m = Matrix(2, 2, [x**2 + y, y**2 + x, 0, x + y]) >>> m Matrix([ [x**2 + y, x + y**2], [ 0, x + y]]) >>> m.is_lower False See Also ======== is_upper is_diagonal is_lower_hessenberg (R­(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pytis_lowers+cC s|j|jkS(s/Checks if a matrix is square. A matrix is square if the number of rows equals the number of columns. The empty matrix is square by definition, since the number of rows and the number of columns are both zero. Examples ======== >>> from sympy import Matrix >>> a = Matrix([[1, 2, 3], [4, 5, 6]]) >>> b = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) >>> c = Matrix([]) >>> a.is_square False >>> b.is_square True >>> c.is_square True (R(R)(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pyR·EscC s |jƒS(süChecks if any elements contain Symbols. Examples ======== >>> from sympy.matrices import Matrix >>> from sympy.abc import x, y >>> M = Matrix([[x, y], [1, 0]]) >>> M.is_symbolic() True (R®(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pyt is_symbolic]s cC sD|}t|ƒs*|rtnd„}n|js7tS|j|ƒS(s‘Check if matrix is symmetric matrix, that is square matrix and is equal to its transpose. By default, simplifications occur before testing symmetry. They can be skipped using 'simplify=False'; while speeding things a bit, this may however induce false negatives. Examples ======== >>> from sympy import Matrix >>> m = Matrix(2, 2, [0, 1, 1, 2]) >>> m Matrix([ [0, 1], [1, 2]]) >>> m.is_symmetric() True >>> m = Matrix(2, 2, [0, 1, 2, 0]) >>> m Matrix([ [0, 1], [2, 0]]) >>> m.is_symmetric() False >>> m = Matrix(2, 3, [0, 0, 0, 0, 0, 0]) >>> m Matrix([ [0, 0, 0], [0, 0, 0]]) >>> m.is_symmetric() False >>> from sympy.abc import x, y >>> m = Matrix(3, 3, [1, x**2 + 2*x + 1, y, (x + 1)**2 , 2, 0, y, 0, 3]) >>> m Matrix([ [ 1, x**2 + 2*x + 1, y], [(x + 1)**2, 2, 0], [ y, 0, 3]]) >>> m.is_symmetric() True If the matrix is already simplified, you may speed-up is_symmetric() test by using 'simplify=False'. >>> bool(m.is_symmetric(simplify=False)) False >>> m1 = m.expand() >>> m1.is_symmetric(simplify=False) True cS s|S(N((R¶((s4lib/python2.7/site-packages/sympy/matrices/common.pyR4¥R5(RR+R·RR¯(R!RR¤((s4lib/python2.7/site-packages/sympy/matrices/common.pyt is_symmetricls 7  cC s |jƒS(s1Checks if the matrix is the upper-Hessenberg form. The upper hessenberg matrix has zero entries below the first subdiagonal. Examples ======== >>> from sympy.matrices import Matrix >>> a = Matrix([[1, 4, 2, 3], [3, 4, 1, 7], [0, 2, 3, 4], [0, 0, 1, 3]]) >>> a Matrix([ [1, 4, 2, 3], [3, 4, 1, 7], [0, 2, 3, 4], [0, 0, 1, 3]]) >>> a.is_upper_hessenberg True See Also ======== is_lower_hessenberg is_upper (R²(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pytis_upper_hessenberg¬sc s&t‡fd†tdˆjƒDƒƒS(s%Check if matrix is an upper triangular matrix. True can be returned even if the matrix is not square. Examples ======== >>> from sympy import Matrix >>> m = Matrix(2, 2, [1, 0, 0, 1]) >>> m Matrix([ [1, 0], [0, 1]]) >>> m.is_upper True >>> m = Matrix(4, 3, [5, 1, 9, 0, 4 , 6, 0, 0, 5, 0, 0, 0]) >>> m Matrix([ [5, 1, 9], [0, 4, 6], [0, 0, 5], [0, 0, 0]]) >>> m.is_upper True >>> m = Matrix(2, 3, [4, 2, 5, 6, 1, 1]) >>> m Matrix([ [4, 2, 5], [6, 1, 1]]) >>> m.is_upper False See Also ======== is_lower is_diagonal is_upper_hessenberg c3 sA|]7}tt|ˆjƒƒD]}ˆ||fjVqqdS(N(R R±R)R£(R9R.R/(R!(s4lib/python2.7/site-packages/sympy/matrices/common.pys ósi(R\R R((R!((R!s4lib/python2.7/site-packages/sympy/matrices/common.pytis_upperÉs*cC s |jƒS(s Checks if a matrix is a zero matrix. A matrix is zero if every element is zero. A matrix need not be square to be considered zero. The empty matrix is zero by the principle of vacuous truth. For a matrix that may or may not be zero (e.g. contains a symbol), this will be None Examples ======== >>> from sympy import Matrix, zeros >>> from sympy.abc import x >>> a = Matrix([[0, 0], [0, 0]]) >>> b = zeros(3, 4) >>> c = Matrix([[0, 1], [0, 0]]) >>> d = Matrix([]) >>> e = Matrix([[x, 0], [0, 0]]) >>> a.is_zero True >>> b.is_zero True >>> c.is_zero False >>> d.is_zero True >>> e.is_zero (R°(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pyR£÷scC s |jƒS(sReturn non-zero values of self.(R³(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pytvaluess(#RRRR›RžR¢R¦R§R©R«R¬R­R®R¯R°R²R³R˜RmRœRŸR¥R¸R¹RºR»R¼R½R·R¾R¿RÀRÁR£RÂ(((s4lib/python2.7/site-packages/sympy/matrices/common.pyR•Øs@               G +%-  @.tMatrixOperationsc B sveZdZd„Zd„Zd„Zd„Zd„Zd„Zd„Z d„Z d „Z d „Z d „Z d „Zd „Zd!d„Zed!eeeeeed„Zed„ƒZddd„Zdd„Zdd„Zed„Zed„Zdeeed„Zd„Zd„Zd„Z ee d!d!dƒZ!eed!d!dƒZ"eZ#d„Z$eZ%d „Z&RS("saProvides basic matrix shape and elementwise operations. Should not be instantiated directly.cC s|jƒjƒS(N(t transposeR¨(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pyt _eval_adjointscC s8|j|j|jg|D]}||ƒ^qƒ}|S(N(R R(R)(R!tfR¶tout((s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_applyfunc"s4cC s2ddlm}m}|j|ƒ|j|ƒfS(Niÿÿÿÿ(tretim(t$sympy.functions.elementary.complexesRÉRÊt applyfunc(R!RÉRÊ((s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_as_real_imag&scC s|jd„ƒS(NcS s |jƒS(N(R¨(R¶((s4lib/python2.7/site-packages/sympy/matrices/common.pyR4,R5(RÌ(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_conjugate+sc s7t|ƒ‰‡‡fd†}ˆjˆjˆj|ƒS(Nc sˆ|ˆ|fS(N((R.R/(tmappingR!(s4lib/python2.7/site-packages/sympy/matrices/common.pyR12s(R<R R(R)(R!tpermR1((RÏR!s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_permute_cols.s c s7t|ƒ‰‡‡fd†}ˆjˆjˆj|ƒS(Nc sˆˆ||fS(N((R.R/(RÏR!(s4lib/python2.7/site-packages/sympy/matrices/common.pyR1;s(R<R R(R)(R!RÐR1((RÏR!s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_permute_rows7s c s#t‡fd†tˆjƒDƒƒS(Nc3 s|]}ˆ||fVqdS(N((R9R.(R!(s4lib/python2.7/site-packages/sympy/matrices/common.pys As(R‚R R((R!((R!s4lib/python2.7/site-packages/sympy/matrices/common.pyt _eval_trace@sc s"ˆjˆjˆj‡fd†ƒS(Nc sˆ||fS(N((R.R/(R!(s4lib/python2.7/site-packages/sympy/matrices/common.pyR4DR5(R R)R((R!((R!s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_transposeCscC s |jƒS(s-Conjugate transpose or Hermitian conjugation.(RÅ(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pytadjointFscC s(t|ƒstdƒ‚n|j|ƒS(sLApply a function to each element of the matrix. Examples ======== >>> from sympy import Matrix >>> m = Matrix(2, 2, lambda i, j: i*2+j) >>> m Matrix([ [0, 1], [2, 3]]) >>> m.applyfunc(lambda i: 2*i) Matrix([ [0, 2], [4, 6]]) s`f` must be callable.(tcallableR[RÈ(R!RÆ((s4lib/python2.7/site-packages/sympy/matrices/common.pyRÌJs cC s |jƒS(s@Returns a tuple containing the (real, imaginary) part of matrix.(RÍ(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pyt as_real_imagascC s |jƒS(s$Return the by-element conjugation. Examples ======== >>> from sympy.matrices import SparseMatrix >>> from sympy import I >>> a = SparseMatrix(((1, 2 + I), (3, 4), (I, -I))) >>> a Matrix([ [1, 2 + I], [3, 4], [I, -I]]) >>> a.C Matrix([ [ 1, 2 - I], [ 3, 4], [-I, I]]) See Also ======== transpose: Matrix transposition H: Hermite conjugation D: Dirac conjugation (RÎ(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pyR¨escK s|jd„ƒS(NcS s |jƒS(N(tdoit(R¶((s4lib/python2.7/site-packages/sympy/matrices/common.pyR4ƒR5(RÌ(R!R((s4lib/python2.7/site-packages/sympy/matrices/common.pyRØ‚sc s|j‡‡fd†ƒS(s&Apply evalf() to each element of self.c s|jˆˆS(N(tevalf(R.(toptionstprec(s4lib/python2.7/site-packages/sympy/matrices/common.pyR4‡R5(RÌ(R!RÛRÚ((RÚRÛs4lib/python2.7/site-packages/sympy/matrices/common.pyRÙ…sc s.|j‡‡‡‡‡‡‡‡‡f d†ƒS(s-Apply core.function.expand to each entry of the matrix. Examples ======== >>> from sympy.abc import x >>> from sympy.matrices import Matrix >>> Matrix(1, 1, [x*(x+1)]) Matrix([[x*(x + 1)]]) >>> _.expand() Matrix([[x**2 + x]]) c  s%|jˆˆˆˆˆˆˆˆˆS(N(texpand(R¶( tbasictdeepthintstlogtmodulustmult multinomialt power_baset power_exp(s4lib/python2.7/site-packages/sympy/matrices/common.pyR4˜s(RÌ( R!RÞRáRäRåRâRàRãRÝRß(( RÝRÞRßRàRáRâRãRäRås4lib/python2.7/site-packages/sympy/matrices/common.pyR܉scC s |jjS(s™Return Hermite conjugate. Examples ======== >>> from sympy import Matrix, I >>> m = Matrix((0, 1 + I, 2, 3)) >>> m Matrix([ [ 0], [1 + I], [ 2], [ 3]]) >>> m.H Matrix([[0, 1 - I, 2, 3]]) See Also ======== conjugate: By-element conjugation D: Dirac conjugation (tTtC(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pytHœsR(tforwardc sØ|dkrd}n|dkr*d}n|dkr?d}n|dkrctdj|ƒƒ‚n|dkr‡td j|ƒƒ‚n|dkrœ|jn|j‰t‡fd †tt|ƒƒDƒƒsÜtd ƒ‚ny‡t|d ƒd ksût ‚|dkrt |ƒ}ntt ˆƒƒ}x1|D])\}}||||||<||>> from sympy.matrices import eye >>> M = eye(3) >>> M.permute([[0, 1], [0, 2]], orientation='rows', direction='forward') Matrix([ [0, 0, 1], [1, 0, 0], [0, 1, 0]]) >>> from sympy.matrices import eye >>> M = eye(3) >>> M.permute([[0, 1], [0, 2]], orientation='rows', direction='backward') Matrix([ [0, 1, 0], [0, 0, 1], [1, 0, 0]]) tforwardsRét backwardstbackwardtcolumnsR)s?direction='{}' is an invalid kwarg. Try 'forward' or 'backward'R(s:orientation='{}' is an invalid kwarg. Try 'rows' or 'cols'c3 s+|]!}d|ko ˆknVqdS(iN((R9R´(t max_index(s4lib/python2.7/site-packages/sympy/matrices/common.pys êss`swap` indices out of range.iiiÿÿÿÿ(t PermutationR~N(RéRì(R(R)(R[RTR(R)R\RR<t IndexErrorR=tAssertionErrortreversedR tsympy.combinatoricsRïRÒRÑ(R!RÐt orientationt directionRÏR.R/Rï((Rîs4lib/python2.7/site-packages/sympy/matrices/common.pytpermute¶s>$          ( !    cC s|j|ddd|ƒS(s…Alias for `self.permute(swaps, orientation='cols', direction=direction)` See Also ======== permute RôR)Rõ(Rö(R!tswapsRõ((s4lib/python2.7/site-packages/sympy/matrices/common.pyt permute_colsscC s|j|ddd|ƒS(s…Alias for `self.permute(swaps, orientation='rows', direction=direction)` See Also ======== permute RôR(Rõ(Rö(R!R÷Rõ((s4lib/python2.7/site-packages/sympy/matrices/common.pyt permute_rowssc s|j‡fd†ƒS(s²Apply refine to each element of the matrix. Examples ======== >>> from sympy import Symbol, Matrix, Abs, sqrt, Q >>> x = Symbol('x') >>> Matrix([[Abs(x)**2, sqrt(x**2)],[sqrt(x**2), Abs(x)**2]]) Matrix([ [ Abs(x)**2, sqrt(x**2)], [sqrt(x**2), Abs(x)**2]]) >>> _.refine(Q.real(x)) Matrix([ [ x**2, Abs(x)], [Abs(x), x**2]]) c s t|ˆƒS(N(R(R¶(t assumptions(s4lib/python2.7/site-packages/sympy/matrices/common.pyR4+R5(RÌ(R!Rú((Rús4lib/python2.7/site-packages/sympy/matrices/common.pyRsc s|j‡‡‡fd†ƒS(s«Replaces Function F in Matrix entries with Function G. Examples ======== >>> from sympy import symbols, Function, Matrix >>> F, G = symbols('F, G', cls=Function) >>> M = Matrix(2, 2, lambda i, j: F(i+j)) ; M Matrix([ [F(0), F(1)], [F(1), F(2)]]) >>> N = M.replace(F,G) >>> N Matrix([ [G(0), G(1)], [G(1), G(2)]]) c s|jˆˆˆƒS(N(treplace(R¶(tFtGtmap(s4lib/python2.7/site-packages/sympy/matrices/common.pyR4?R5(RÌ(R!RüRýRþ((RüRýRþs4lib/python2.7/site-packages/sympy/matrices/common.pyRû-sg333333û?c s|j‡‡‡‡fd†ƒS(suApply simplify to each element of the matrix. Examples ======== >>> from sympy.abc import x, y >>> from sympy import sin, cos >>> from sympy.matrices import SparseMatrix >>> SparseMatrix(1, 1, [x*sin(y)**2 + x*cos(y)**2]) Matrix([[x*sin(y)**2 + x*cos(y)**2]]) >>> _.simplify() Matrix([[x]]) c  s"|jdˆdˆdˆdˆƒS(Ntratiotmeasuretrationaltinverse(R(R¶(RRRÿR(s4lib/python2.7/site-packages/sympy/matrices/common.pyR4Os(RÌ(R!RÿRRR((RRRÿRs4lib/python2.7/site-packages/sympy/matrices/common.pyRAsc s|j‡‡fd†ƒS(sdReturn a new matrix with subs applied to each entry. Examples ======== >>> from sympy.abc import x, y >>> from sympy.matrices import SparseMatrix, Matrix >>> SparseMatrix(1, 1, [x]) Matrix([[x]]) >>> _.subs(x, y) Matrix([[y]]) >>> Matrix(_).subs(y, x) Matrix([[x]]) c s|jˆˆŽS(N(tsubs(R¶(RR(s4lib/python2.7/site-packages/sympy/matrices/common.pyR4aR5(RÌ(R!RR((RRs4lib/python2.7/site-packages/sympy/matrices/common.pyRRscC s(|j|jkrtƒ‚n|jƒS(sú Returns the trace of a square matrix i.e. the sum of the diagonal elements. Examples ======== >>> from sympy import Matrix >>> A = Matrix(2, 2, [1, 2, 3, 4]) >>> A.trace() 5 (R(R)RRÓ(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pyttracecs cC s |jƒS(sa Returns the transpose of the matrix. Examples ======== >>> from sympy import Matrix >>> A = Matrix(2, 2, [1, 2, 3, 4]) >>> A.transpose() Matrix([ [1, 3], [2, 4]]) >>> from sympy import Matrix, I >>> m=Matrix(((1, 2+I), (3, 4))) >>> m Matrix([ [1, 2 + I], [3, 4]]) >>> m.transpose() Matrix([ [ 1, 3], [2 + I, 4]]) >>> m.T == m.transpose() True See Also ======== conjugate: By-element conjugation (RÔ(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pyRÄus!sMatrix transposition.sBy-element conjugation.c s|j‡fd†ƒS(stReturn a new matrix with xreplace applied to each entry. Examples ======== >>> from sympy.abc import x, y >>> from sympy.matrices import SparseMatrix, Matrix >>> SparseMatrix(1, 1, [x]) Matrix([[x]]) >>> _.xreplace({x: y}) Matrix([[y]]) >>> Matrix(_).xreplace({y: x}) Matrix([[x]]) c s |jˆƒS(N(txreplace(R¶(trule(s4lib/python2.7/site-packages/sympy/matrices/common.pyR4­R5(RÌ(R!R((Rs4lib/python2.7/site-packages/sympy/matrices/common.pyRžsc s)ddlm‰|j‡‡fd†ƒS(Niÿÿÿÿ(ttrigsimpc s ˆ|ˆS(N((R¶(toptsR(s4lib/python2.7/site-packages/sympy/matrices/common.pyR4³R5(tsympy.simplifyRRÌ(R!R((RRs4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_trigsimp±sN('RRRRÅRÈRÍRÎRÑRÒRÓRÔRÕRÌR×R¨RØR'RÙR¥RÜRmRèRöRøRùRRRûR RRRRÄRæRçRPRt_eval_simplifyR (((s4lib/python2.7/site-packages/sympy/matrices/common.pyRÃsB             O     # tMatrixArithmeticcB sgeZdZdZd„Zd„Zd„Zd„Zd„Zd„Z d„Z d „Z d „Z d „Z ed ƒd „ƒZedƒd„ƒZedƒd„ƒZedƒd„ƒZd„Zedƒd„ƒZedƒd„ƒZedƒd„ƒZedƒd„ƒZedƒd„ƒZedƒd „ƒZed!ƒd"„ƒZd#„ZRS($sUProvides basic matrix arithmetic operations. Should not be instantiated directly.g…ëQ¸$@c s"ˆjˆjˆj‡fd†ƒS(Nc stˆ||fƒS(N(R(R.R/(R!(s4lib/python2.7/site-packages/sympy/matrices/common.pyR4½R5(R R(R)(R!((R!s4lib/python2.7/site-packages/sympy/matrices/common.pyt _eval_Abs¼sc s%ˆjˆjˆj‡‡fd†ƒS(Nc sˆ||fˆ||fS(N((R.R/(R"R!(s4lib/python2.7/site-packages/sympy/matrices/common.pyR4ÁR5(R R(R)(R!R"((R"R!s4lib/python2.7/site-packages/sympy/matrices/common.pyt _eval_add¿sc s+‡‡fd†}ˆjˆjˆj|ƒS(Nc s¦y0t‡‡‡‡fd†tˆjƒDƒƒSWnotk r¡ˆˆdfˆdˆf}x<tdˆjƒD](}|ˆˆ|fˆ|ˆf7}qqW|SXdS(Nc3 s-|]#}ˆˆ|fˆ|ˆfVqdS(N((R9Ra(R.R/R"R!(s4lib/python2.7/site-packages/sympy/matrices/common.pys Æsii(R‚R R)R[(R.R/tretRa(R"R!(R.R/s4lib/python2.7/site-packages/sympy/matrices/common.pyR1Äs0 &(R R(R)(R!R"R1((R"R!s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_matrix_mulÃsc s%ˆjˆjˆj‡‡fd†ƒS(Nc sˆ||fˆ||fS(N((R.R/(R"R!(s4lib/python2.7/site-packages/sympy/matrices/common.pyR4ÕR5(R R(R)(R!R"((R"R!s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_matrix_mul_elementwiseÔsc s+‡‡fd†}ˆjˆjˆj|ƒS(Nc s,t‡‡‡‡fd†tˆjƒDƒƒS(Nc3 s-|]#}ˆˆ|fˆ|ˆfVqdS(N((R9Ra(R.R/R"R!(s4lib/python2.7/site-packages/sympy/matrices/common.pys Ùs(R‚R R)(R.R/(R"R!(R.R/s4lib/python2.7/site-packages/sympy/matrices/common.pyR1Øs(R R(R)(R!R"R1((R"R!s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_matrix_rmul×scC sP|dkr|S|ddkr5||j|dƒS|j|dƒ}||S(Nii(t_eval_pow_by_recursion(R!tnumR((s4lib/python2.7/site-packages/sympy/matrices/common.pyRÜs  c s%ˆjˆjˆj‡‡fd†ƒS(Nc sˆ||fˆS(N((R.R/(R"R!(s4lib/python2.7/site-packages/sympy/matrices/common.pyR4åR5(R R(R)(R!R"((R"R!s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_scalar_muläsc s%ˆjˆjˆj‡‡fd†ƒS(Nc sˆˆ||fS(N((R.R/(R"R!(s4lib/python2.7/site-packages/sympy/matrices/common.pyR4èR5(R R(R)(R!R"((R"R!s4lib/python2.7/site-packages/sympy/matrices/common.pyt_eval_scalar_rmulçsc s8ddlm‰ˆjˆjˆj‡‡‡fd†ƒS(Niÿÿÿÿ(tModc sˆˆ||fˆƒS(N((R.R/(RR"R!(s4lib/python2.7/site-packages/sympy/matrices/common.pyR4ìR5(tsympyRR R(R)(R!R"((RR"R!s4lib/python2.7/site-packages/sympy/matrices/common.pyt _eval_ModêscC s |jƒS(s5Returns a new matrix with entry-wise absolute values.(R (R!((s4lib/python2.7/site-packages/sympy/matrices/common.pyt__abs__ïst__radd__cC sët|ƒ}t|dƒrO|j|jkrOtd|j|jfƒ‚qOnt|dtƒr£||}}|jt||ƒkr–||}}n|j|ƒSt|dtƒrÅt j||ƒSt dt |ƒt |ƒfƒ‚dS(s>Return self + other, raising ShapeError if shapes don't match.R*sMatrix size mismatch: %s + %sR{t is_MatrixLikescannot add %s and %sN( t _matrixifyR|R*RR€Rt __class__R7RR R[RV(R!R"R tb((s4lib/python2.7/site-packages/sympy/matrices/common.pyt__add__ós   t__rdiv__cC s|tj|S(N(RRq(R!R"((s4lib/python2.7/site-packages/sympy/matrices/common.pyt__div__ st __rmatmul__cC sCt|ƒ}t|dtƒ r6t|dtƒ r6tS|j|ƒS(NR{R(RR€RtNotImplementedt__mul__(R!R"((s4lib/python2.7/site-packages/sympy/matrices/common.pyt __matmul__s &t__rmul__cC sèt|ƒ}t|dƒrlt|jƒdkrl|jd|jdkrltd|j|jfƒ‚qlnt|dtƒr‹|j|ƒSt|dtƒr­tj||ƒSt |t ƒsäy|j |ƒSWqät k ràqäXnt S(sReturn self*other where other is either a scalar or a matrix of compatible dimensions. Examples ======== >>> from sympy.matrices import Matrix >>> A = Matrix([[1, 2, 3], [4, 5, 6]]) >>> 2*A == A*2 == Matrix([[2, 4, 6], [8, 10, 12]]) True >>> B = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) >>> A*B Matrix([ [30, 36, 42], [66, 81, 96]]) >>> B*A Traceback (most recent call last): ... ShapeError: Matrices size mismatch. >>> See Also ======== matrix_multiply_elementwise R*iiisMatrix size mismatch: %s * %s.R{R(RR|R=R*RR€RRR RYRRR[R$(R!R"((s4lib/python2.7/site-packages/sympy/matrices/common.pyR%s $  cC s |jdƒS(Niÿÿÿÿ(R(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pyt__neg__Kst__rpow__cC sŸ|j|jkrtƒ‚n|}t|ddƒ}t|ƒ}|jr(|ddkr(|jdkr…|j|d|ggƒS|dkr­|j|j|jd„ƒS|dkrÏ| }|jƒ}nL|jdkr|dkr|dk ry||ƒSWqt k rqXn|j |ƒS|j rp|j dkrp|j ƒdkrpddl m}|||ƒSt|ttfƒr||ƒStd ƒ‚dS( Nt_matrix_pow_by_jordan_blocksiicS st||kƒS(N(tint(R.R/((s4lib/python2.7/site-packages/sympy/matrices/common.pyR4YR5ii †iÿÿÿÿ(tMatPowsIOnly SymPy expressions or integers are supported as exponent for matrices(R(R)RR€R'Rt is_NumberR tinvRRt is_negativetdettsympy.matrices.expressionsR,RYR tfloatR[(R!RR t jordan_powR,((s4lib/python2.7/site-packages/sympy/matrices/common.pyt__pow__Ns4    '  +  R cC s||S(N((R!R"((s4lib/python2.7/site-packages/sympy/matrices/common.pyRosR&cC sCt|ƒ}t|dtƒ r6t|dtƒ r6tS|j|ƒS(NR{R(RR€RR$R'(R!R"((s4lib/python2.7/site-packages/sympy/matrices/common.pyR#ss &R%cC sât|ƒ}t|dƒr\t|jƒdkr\|jd|jdkr\tdƒ‚q\nt|dtƒr…|j|jƒ|ƒSt|dtƒr§t j ||ƒSt |t ƒsÞy|j |ƒSWqÞtk rÚqÞXntS(NR*iiisMatrix size mismatch.R{R(RR|R=R*RR€RR t as_mutableR RRYRRR[R$(R!R"((s4lib/python2.7/site-packages/sympy/matrices/common.pyR'{s $ t__sub__cC s | |S(N((R!R ((s4lib/python2.7/site-packages/sympy/matrices/common.pyt__rsub__”sR7cC s || S(N((R!R ((s4lib/python2.7/site-packages/sympy/matrices/common.pyR6˜st __rtruediv__cC s |j|ƒS(N(R"(R!R"((s4lib/python2.7/site-packages/sympy/matrices/common.pyt __truediv__œscC s@|j|jkr3tdj|j|jƒƒ‚n|j|ƒS(s²Return the Hadamard product (elementwise product) of A and B Examples ======== >>> from sympy.matrices import Matrix >>> A = Matrix([[0, 1, 2], [3, 4, 5]]) >>> B = Matrix([[1, 10, 100], [100, 10, 1]]) >>> A.multiply_elementwise(B) Matrix([ [ 0, 10, 200], [300, 40, 5]]) See Also ======== cross dot multiply s!Matrix shapes must agree {} != {}(R*RRTR(R!R"((s4lib/python2.7/site-packages/sympy/matrices/common.pytmultiply_elementwise s!(RRRt _op_priorityR RRRRRRRRRR R R"R&R%R(R4RR#R'R7R6R9R:(((s4lib/python2.7/site-packages/sympy/matrices/common.pyR ¶s2          4 !t MatrixCommoncB seZdZeZRS(ssAll common matrix operations including basic arithmetic, shaping, and special matrices like `zeros`, and `eye`.(RRRR¥t _diff_wrt(((s4lib/python2.7/site-packages/sympy/matrices/common.pyR<»st_MinimalMatrixcB sƒeZdZeZeeƒZdZeZ e Z e d„ƒZ d d d„Zd„Zd„Zd„Zd„Zed„ƒZRS( sûClass providing the minimum functionality for a matrix-like object and implementing every method required for a `MatrixRequired`. This class does not have everything needed to become a full-fledged sympy object, but it will satisfy the requirements of anything inheriting from `MatrixRequired`. If you wish to make a specialized matrix type, make sure to implement these methods and properties with the exception of `__init__` and `__repr__` which are included for convenience.icO s |||ŽS(N((RRR((s4lib/python2.7/site-packages/sympy/matrices/common.pyR Ósc sWtˆƒr4t‡‡fd†t|ƒDƒƒ‰nˆdkrUˆdkrU|‰ntˆd|ˆfƒ\}‰ygˆdkr—ˆdkr—|‰ntˆdƒ‰tˆƒ}gˆD]}|D] }|^qÄqº‰Wnttfk rónXt‡fd†ˆDƒƒˆ_ |ˆˆ_ ˆ_ ˆj dksDˆj dkrSt dƒ‚ndS(Nc3 s1|]'}tˆƒD]}ˆ||ƒVqqdS(N(R (R9R.R/(R)R;(s4lib/python2.7/site-packages/sympy/matrices/common.pys ÚsR*ic3 s|]}ˆj|ƒVqdS(N(t_sympify(R9R¶(R!(s4lib/python2.7/site-packages/sympy/matrices/common.pys çss.Cannot initialize matrix with given parameters( RR<R R'R€R=RðR[RµR;R(R)R(R!R(R)R;tlR¶((R)R;R!s4lib/python2.7/site-packages/sympy/matrices/common.pyt__init__×s" (   *c s ‡fd†}‡fd†}t|tƒrÿ|\}}t|tƒsWt|tƒrí|||ƒ\}}ttˆjƒƒ|ttˆjƒƒ|}‰‡‡fd†|Dƒ}ˆjt|ƒtˆƒt‡fd†|DƒƒƒS|||ƒ}nˆj |S(Nc sŠt|tƒs(t||ddƒ}nt|jˆjƒŒ}t|tƒsht||ddƒ}nt|jˆjƒŒ}||fS(s“Ensure that row_slice and col_slice don't have `None` in their arguments. Any integers are converted to slices of length 1iN(RYtsliceR'R?R(R)(t row_slicet col_slice(R!(s4lib/python2.7/site-packages/sympy/matrices/common.pyt_normalize_slicesísc s|ˆj|S(sXReturn the index in _mat corresponding to the (i,j) position in the matrix. (R)(R.R/(R!(s4lib/python2.7/site-packages/sympy/matrices/common.pyt_coord_to_indexûsc3 s-|]#}ˆD]}|ˆj|Vq qdS(N(R)(R9R.R/(R:R!(s4lib/python2.7/site-packages/sympy/matrices/common.pys sc3 s|]}ˆj|VqdS(N(R;(R9R.(R!(s4lib/python2.7/site-packages/sympy/matrices/common.pys s( RYRµRBR<R R(R)R R=R;(R!R$RERFR.R/R>R?((R:R!s4lib/python2.7/site-packages/sympy/matrices/common.pyR%ìs cC sNyt||ƒWntk r%tSX|j|jkoMt|ƒt|ƒkS(N(R7R[RR*R<(R!R"((s4lib/python2.7/site-packages/sympy/matrices/common.pyR# s  cC s|j|jS(N(R(R)(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pyR& scC sdj|j|j|jƒS(Ns_MinimalMatrix({}, {}, {})(RTR(R)R;(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pyt__repr__ scC s|j|jfS(N(R(R)(R!((s4lib/python2.7/site-packages/sympy/matrices/common.pyR*" sN(RRRR¥Rt staticmethodRR?t_class_priorityR{Rt is_MatrixExprR,R R'RAR%R#R&RGRmR*(((s4lib/python2.7/site-packages/sympy/matrices/common.pyR>Âs  '   t_MatrixWrappercB s2eZdZeZdd„Zd„Zd„ZRS(s8Wrapper class providing the minimum functionality for a matrix-like object: .rows, .cols, .shape, indexability, and iterability. CommonMatrix math operations should work on matrix-like objects. For example, wrapping a numpy matrix in a MatrixWrapper allows it to be passed to CommonMatrix. cC s4||_|dkr|jn|\|_|_dS(N(R;R'R*R(R)(R!R;R*((s4lib/python2.7/site-packages/sympy/matrices/common.pyRA0 s cC st|j|ƒS(sMMost attribute access is passed straight through to the stored matrix(R€R;(R!tattr((s4lib/python2.7/site-packages/sympy/matrices/common.pyt __getattr__4 scC s|jj|ƒS(N(R;R%(R!R$((s4lib/python2.7/site-packages/sympy/matrices/common.pyR%9 sN( RRRR¥RR'RARMR%(((s4lib/python2.7/site-packages/sympy/matrices/common.pyRK' s   cC sKt|dtƒr|St|dƒrGt|jƒdkrGt|ƒSn|S(s‘If `mat` is a Matrix or is matrix-like, return a Matrix or MatrixWrapper object. Otherwise `mat` is passed through without modification.R{R*i(R€RR|R=R*RK(R;((s4lib/python2.7/site-packages/sympy/matrices/common.pyR= s  cC s²t|ƒtk rRt|ddƒ}|dk r<|ƒ}qRtd|fƒ‚n|dk r¨|dkrw||7}n|dkoŒ||ks¨td|fƒ‚q¨nt|ƒS(s>Return integer after making positive and validating against n.t __index__sInvalid index a[%r]isIndex out of range: a[%s]N(RVR+R€R'Rð(R/RPtjindex((s4lib/python2.7/site-packages/sympy/matrices/common.pyR^I s     cC sÏt|ddƒ}t|ddƒ}d||fkrY|j|jkrO|jS|jSnyddl}Wntk r|n3Xt||jƒr–|jSt||jƒr¯|jStd|j|jfƒ‚dS(sÊ Get the type of the result when combining matrices of different types. Currently the strategy is that immutability is contagious. Examples ======== >>> from sympy import Matrix, ImmutableMatrix >>> from sympy.matrices.common import classof >>> M = Matrix([[1, 2], [3, 4]]) # a Mutable Matrix >>> IM = ImmutableMatrix([[1, 2], [3, 4]]) >>> classof(M, IM) RIiÿÿÿÿNsIncompatible classes %s, %s( R€R'RIRtnumpyt ImportErrorRYtndarrayR[(tAtBt priority_At priority_BRP((s4lib/python2.7/site-packages/sympy/matrices/common.pyR7Y s  N(:Rt __future__RRt collectionsRtinspectRtsympy.assumptions.refineRtsympy.core.basicRtsympy.core.compatibilityRRRR R tsympy.core.decoratorsR tsympy.core.exprR tsympy.core.functionR tsympy.core.singletonRtsympy.core.symbolRtsympy.core.sympifyRtsympy.functionsRR RR+tsympy.utilities.exceptionsRtsympy.utilities.iterablesRt ExceptionRRSRRtobjectRR-RnR•RÃR R<R>RKRR'R^R7(((s4lib/python2.7/site-packages/sympy/matrices/common.pytsN(ÿÿÿyÿÿEÿœÿ e