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Examples ======== >>> from sympy import MatrixSymbol >>> A = MatrixSymbol('A', 3, 3) >>> y = MatrixSymbol('y', 3, 1) >>> x = (A.T*A).I * A * y See Also ======== MatrixSymbol, MatAdd, MatMul, Transpose, Inverse Fg&@TNÚkindcOs"tt|ƒ}tj|g|¢Ri|¤ŽSr )ÚmaprrÚ__new__)ÚclsÚargsÚkwargsr%r%r&r/Ps zMatrixExpr.__new__)ÚreturncCst‚dSr ©ÚNotImplementedError©Úselfr%r%r&ÚshapeVszMatrixExpr.shapecCstSr ©ÚMatAddr6r%r%r&Ú_add_handlerZszMatrixExpr._add_handlercCstSr ©ÚMatMulr6r%r%r&Ú_mul_handler^szMatrixExpr._mul_handlercCsttj|ƒ ¡Sr )r=rÚNegativeOneÚdoitr6r%r%r&Ú__neg__bszMatrixExpr.__neg__cCst‚dSr r4r6r%r%r&Ú__abs__eszMatrixExpr.__abs__ÚotherÚ__radd__cCst||dd ¡S©NT)Úcheck©r:r@©r7rCr%r%r&Ú__add__hszMatrixExpr.__add__rIcCst||dd ¡SrErGrHr%r%r&rDmszMatrixExpr.__radd__Ú__rsub__cCst||dd ¡SrErGrHr%r%r&Ú__sub__rszMatrixExpr.__sub__rKcCst||dd ¡SrErGrHr%r%r&rJwszMatrixExpr.__rsub__Ú__rmul__cCst||ƒ ¡Sr ©r=r@rHr%r%r&Ú__mul__|szMatrixExpr.__mul__cCst||ƒ ¡Sr rMrHr%r%r&Ú __matmul__szMatrixExpr.__matmul__rNcCst||ƒ ¡Sr rMrHr%r%r&rL†szMatrixExpr.__rmul__cCst||ƒ ¡Sr rMrHr%r%r&Ú__rmatmul__‹szMatrixExpr.__rmatmul__Ú__rpow__cCst||ƒ ¡Sr )ÚMatPowr@rHr%r%r&Ú__pow__szMatrixExpr.__pow__rScCstdƒ‚dS)NzMatrix Power not definedr4rHr%r%r&rQ•szMatrixExpr.__rpow__Ú__rtruediv__cCs||tjSr )rr?rHr%r%r&Ú__truediv__šszMatrixExpr.__truediv__rUcCs tƒ‚dSr r4rHr%r%r&rTŸszMatrixExpr.__rtruediv__cCs |jdS©Nr©r8r6r%r%r&Úrows¥szMatrixExpr.rowscCs |jdS©NérWr6r%r%r&Úcols©szMatrixExpr.colscCs|j|jkSr ©rXr[r6r%r%r&Ú is_squareszMatrixExpr.is_squarecCsddlm}|t|ƒƒS©Nr)ÚAdjoint)Ú"sympy.matrices.expressions.adjointr_Ú Transpose©r7r_r%r%r&Ú_eval_conjugate±szMatrixExpr._eval_conjugatecKs0tj|| ¡}|| ¡dtj}||fS©Né)rZHalfrcZ ImaginaryUnit)r7ÚdeepÚhintsÚrealZimr%r%r&Úas_real_imagµszMatrixExpr.as_real_imagcCst|ƒSr ©ÚInverser6r%r%r&Ú _eval_inverseºszMatrixExpr._eval_inversecCst|ƒSr ©ÚDeterminantr6r%r%r&Ú_eval_determinant½szMatrixExpr._eval_determinantcCst|ƒSr ©rar6r%r%r&Ú_eval_transposeÀszMatrixExpr._eval_transposecCs t||ƒS)zÙ Override this in sub-classes to implement simplification of powers. 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Examples ======== >>> from sympy import Identity >>> I = Identity(3) >>> I I >>> I.as_explicit() Matrix([ [1, 0, 0], [0, 1, 0], [0, 0, 1]]) See Also ======== as_mutable: returns mutable Matrix type z..)Úranger[)rur6)rr&rx\sþÿz*MatrixExpr.as_explicit..)r¦r‡Úsympy.matrices.immutabler¨rªrX)r7r¨r%r6r&Úas_explicit?sÿþzMatrixExpr.as_explicitcCs| ¡ ¡S)a³ Returns a dense, mutable matrix with elements represented explicitly Examples ======== >>> from sympy import Identity >>> I = Identity(3) >>> I I >>> I.shape (3, 3) >>> I.as_mutable() Matrix([ [1, 0, 0], [0, 1, 0], [0, 0, 1]]) See Also ======== as_explicit: returns ImmutableDenseMatrix )r¬Ú as_mutabler6r%r%r&r`szMatrixExpr.as_mutablecCsRddlm}||jtd}t|jƒD](}t|jƒD]}|||f|||f<q2q$|S)Nr)Úempty)Zdtype)Znumpyr®r8ÚobjectrªrXr[)r7r®r!rrŽr%r%r&Ú __array__yszMatrixExpr.__array__cCs| ¡ |¡S)zÅ Test elementwise equality between matrices, potentially of different types >>> from sympy import Identity, eye >>> Identity(3).equals(eye(3)) True )r¬ÚequalsrHr%r%r&r±s zMatrixExpr.equalscCs|Sr r%r6r%r%r&ÚcanonicalizeŒszMatrixExpr.canonicalizecCsdt|ƒfSrYr<r6r%r%r&Ú as_coeff_mmulszMatrixExpr.as_coeff_mmulcCsTddlm}ddlm}g}|dur.| |¡|dur@| |¡|||d}||ƒS)aÎ Parse expression of matrices with explicitly summed indices into a matrix expression without indices, if possible. 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This matrix has a shape and can be included in Matrix Expressions Examples ======== >>> from sympy import MatrixSymbol, Identity >>> A = MatrixSymbol('A', 3, 4) # A 3 by 4 Matrix >>> B = MatrixSymbol('B', 4, 3) # A 4 by 3 Matrix >>> A.shape (3, 4) >>> 2*A*B + Identity(3) I + 2*A*B FTcCsLt|ƒt|ƒ}}| |¡| |¡t|tƒr8t|ƒ}t ||||¡}|Sr )rrŠršrôrrr/)r0rör~r÷rør%r%r&r/›s zMatrixSymbol.__new__cCs|jd|jdfS)NrZrerðr6r%r%r&r8¦szMatrixSymbol.shapecCs|jdjSrV)r1rör6r%r%r&röªszMatrixSymbol.namecKst|||ƒSr )rïrŒr%r%r&r®szMatrixSymbol._entrycCs|hSr r%r6r%r%r&Úfree_symbols±szMatrixSymbol.free_symbolscKs|Sr r%rzr%r%r&r{µszMatrixSymbol._eval_simplifycCst|jd|jdƒS©NrrZ)r‚r8rƒr%r%r&r¸szMatrixSymbol._eval_derivativecCsÀ||krj|jddkr,t|jd|jdƒntj}|jddkrVt|jd|jdƒntj}t||gƒgS|jddkr†t|jdƒntj}|jddkr¨t|jdƒntj}t||gƒgSdSr)r8r‚rrþÚ_LeftRightArgsrçr)r7rvÚfirstÚsecondr%r%r&rì¼s**ÿ""ÿz*MatrixSymbol._eval_derivative_matrix_linesN)r‹r¿rÀrÁrÅrÆrr/rÈr8rörrr{rrìr%r%r%r&r†s rcCsdd„|jDƒS)NcSsg|]}|jr|‘qSr%)rÂ)ruÚsymr%r%r&rxÌryz"matrix_symbols..)r©rºr%r%r&Úmatrix_symbolsËsr c@sžeZdZdZejfdd„Zedd„ƒZej dd„ƒZedd„ƒZ e j d d„ƒZ d d„Zdd „Ze dd„ƒZdd„Zdd„Zdd„Zdd„Zdd„Zdd„ZdS)ra‘ Helper class to compute matrix derivatives. 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