# Symbolic matrices

 4 Hi, is it possible to define totally symbolic matrices. Something like N,M = var('N,M') a = matrix(SR, N, M) b = matrix(SR, M, N) c = a.dot(b)  So that c[i,j] = sum(a[i,k]*b[k,i],k,1,M) or do you really always need an non-symblic matrix size? If you really always need a non-symblic matrix type, is it possible to define the symbolic variables automatically? Something like a = matrix(SR, 2, 3) print a[i,j] output: a_(i,j)  Thanks! Manuel asked Apr 12 '11 Manuels 43 ● 1 ● 2 ● 5 niles 3725 ● 7 ● 45 ● 104 http://nilesjohnson.net/

 3 There is the Tensor module in sympy. If that is the type of thing that you're looking for (ie explicit indices). The was some code put on the sage devel group last year: abstract matrices, which allows basic manipulation of abstract matrices and vectors. It's not complete, but it does the basics and should be easy to extend if you need more. As far as I can tell, it hasn't been worked on since the original postings. I hope it's ok if I post the code here: from sage.structure.element import Element from sage.combinat.free_module import CombinatorialFreeModule # TODO: doc and tests for all of those class SymbolicMatrix(SageObject): def __init__(self, name, nrows, ncols, inverted = False, transposed = False): #Element.__init__(self, parent) self._name = name self._nrows = nrows self._ncols = ncols self._inverted = inverted self._transposed = transposed def _repr_(self): result = self._name if self._inverted: result += "^-1" if self._transposed: result += "^t" return result def transpose(self): result = copy(self) result._transposed = not self._transposed (result._nrows, result._ncols) = (self._ncols, self._nrows) return result def __invert__(self): assert self._nrows == self._ncols, "Can't inverse non square matrix" result = copy(self) result._inverted = not self._inverted return result class SymbolicMatrixAlgebra(CombinatorialFreeModule): r""" EXAMPLES:: sage: Alg = SymbolicMatrixAlgebra(QQ) sage: A = Alg.matrix("A",3,2) sage: B = Alg.matrix("B",2,3) sage: C = Alg.matrix("C",2,3) sage: D = Alg.matrix("D",3,3) Example 1: Adding/Multiplying matrices of correct size:: sage: A * (B + C) A B + A C Example 2: Transposing a sum of matrices:: sage: (B + C).transpose() C^t + B^t Example 3: Transposing a product of matrices:: sage: (A * B).transpose() B^t A^t Example 4: Inverting a product of matrices:: sage: (A * B)^-1 B^-1 A^-1 Example 5: Multiplying by its inverse:: sage: D * D^-1 # todo: not implemented I TODO: decide on the best output; do we want to be able to copy-paste back? do we prefer short notations? .. warnings:: The identity does not know it is size, so the following should complain, but does not:: sage: D * D^-1 * A TODO: describe all the abuses """ def __init__(self, R): """ EXAMPLES:: sage: A = AlgebrasWithBasis(QQ).example(); A An example of an algebra with basis: the free algebra on the generators ('a', 'b', 'c') over Rational Field sage: TestSuite(A).run() """ CombinatorialFreeModule.__init__(self, R, Words(), category = AlgebrasWithBasis(R)) def matrix(self, name, nrows, ncols): """ TODO: doctest""" return self.monomial(Word([SymbolicMatrix(name, nrows, ncols)])) def _repr_(self): """ EXAMPLES:: sage: SymbolicMatrixAlgebra(QQ) The symbolic matrix algebra over Rational Field """ return "The symbolic matrix algebra over %s"%(self.base_ring()) @cached_method def one_basis(self): """ Returns the empty word, which index the one of this algebra, as per :meth:AlgebrasWithBasis.ParentMethods.one_basis. EXAMPLES:: sage: Alg = SymbolicMatrixAlgebra(QQ) sage: Alg.one_basis() word: sage: A.one() I """ return self.basis().keys()([]) def product_on_basis(self, w1, w2): r""" Product of basis elements, as per :meth:AlgebrasWithBasis.ParentMethods.product_on_basis. EXAMPLES:: sage: Alg = SymbolicMatrixAlgebra(QQ) sage: P = Alg.matrix("P", 3, 2) sage: Q = Alg.matrix("Q", 3, 2) sage: R = Alg.matrix("R", 2, 2) sage: S = Alg.matrix("S", 2, 3) sage: P * P Traceback (most recent call last): ... AssertionError: Non-conformable matrices: matrix sizes are incompatible for multiplication sage: P * R * S P*R*S sage: (P+Q) * R P*R + Q*R sage: (P+Q) * R * S P*R*S + Q*R*S sage: S * (P+Q) * R """ if len(w1) == 0: return self.monomial(w2) if len(w2) == 0: return self.monomial(w1) assert w1[-1]._ncols == w2[0]._nrows, "Non-conformable matrices: matrix sizes are incompatible for multiplication" # TODO: handle cancelations between w1[-1] and w2[0] return self.monomial(w1 + w2) # TODO: define an_element def _repr_term(self, t): """ EXAMPLES:: sage: Alg.one() # indirect doctest I sage: Alg.an_element() # todo: not implemented """ if len(t) == 0: return 'I' else: return ' '.join(repr(m) for m in t) class Element(CombinatorialFreeModule.Element): def transpose(self): """ EXAMPLES:: sage: Alg = SymbolicMatrixAlgebra(QQ) sage: A = Alg.matrix("A", 3, 2) sage: B = Alg.matrix("B", 3, 2) sage: C = Alg.matrix("C", 2, 2) sage: D = Alg.matrix("D", 2, 3) sage: x = D * (A+B) * C sage: x D B C + D A C sage: x.transpose() C^t B^t D^t + C^t A^t D^t """ return self.map_support(lambda w: Word(m.transpose() for m in reversed(w))) def __invert__(self): """ EXAMPLES:: sage: Alg = SymbolicMatrixAlgebra(QQ) sage: A = Alg.matrix("A", 2, 2) sage: B = Alg.matrix("B", 2, 2) sage: C = Alg.matrix("C", 2, 2) sage: D = Alg.matrix("D", 3, 2) sage: E = Alg.matrix("E", 2, 3) sage: ~(A*B*C) C^-1 B^-1 A^-1 sage: ~(A*B^-1*C^-1) C B A^-1 sage: ~(D*C*E) Traceback (most recent call last): ... AssertionError: Can't inverse non square matrix sage: ~(A+B) Traceback (most recent call last): ... AssertionError: Can't inverse non trivial linear combinations of matrices """ assert len(self) == 1, "Can't inverse non trivial linear combinations of matrices" (w, c) = self.leading_item() return self.parent().term(Word(~m for m in reversed(w)), c)  posted Apr 12 '11 Simon 285 ● 2 ● 7 ● 18 http://www.physics.uwa.ed... slelievre 702 ● 10 ● 23 1 That's an interesting thread, which I had forgotten. Did it ever turn into a ticket? kcrisman (Apr 12 '11)@kcrisman - I had a quick look, but couldn't find anything. Simon (Apr 13 '11) Then open one and post the code :) kcrisman (Apr 14 '11)@kcrisman: Remind me again in a couple of months after I've written my thesis - then I'll have a go at completing and posting the code... Simon (Apr 15 '11)
 3 I'm not sure if there's a way to define a symbolic matrix the way you describe above. However, one could create a matrix populated only by distinct symbolic variables. Here's a quick, though supposedly not quickest, way to do so: sage: N = 3 sage: s = join(['a_%d%d' %(i,j) for (i,j) in CartesianProduct(range(N),range(N))]) sage: a = var(s) sage: A = matrix(SR,N,N,a) sage: print A[1,2] a_12  The second line just creates a string with the variable names $a_{ij}$ for $i,j \in \mathbb{Z}_n$ and the third parses the string and creates a list of the symbolic variables. I can now create another matrix $B$ and multiply them: sage: s = join(['b_%d%d' %(i,j) for (i,j) in CartesianProduct(range(N),range(N))]) sage: b = var(s) sage: B = matrix(SR,N,N,b) sage: C = A*B sage: print C[1,2] a_10*b_02 + a_11*b_12 + a_12*b_22  It's not completely automatic, and perhaps not pretty, but I hope it helps. posted Apr 12 '11 cswiercz 839 ● 6 ● 17 ● 33 http://www.cswiercz.info/ This isn't a very "beautiful" method, but I can't think of anything better. :-) Anybody know of a cleaner approach? By the way, I would put an underscore between the indices, so the variables would read: "a_1_0", "b_3_5", and "c_23_829" (just an example). And would this method work for sparse matrices? Kelvin Li (Apr 12 '11) I agree that it's not pretty, but even if Sage had the ability to define these "purely symbolic" matrices one would have to call the elements _something_. Perhaps this sort fo thing would happen in the __init__() method. Also, I use the notation "a_ij" instead of "a_i_j" because when you execute "show(a_ij)", or enable typesetting in the Sage Notebook, it looks prettier. :) cswiercz (Apr 12 '11) Ignoring all else, I would like to reference and name the elements consistently: a[1,2] being printed exactly as "a[1,2]". Then again, I can't ignore the facts. :-) Kelvin Li (Apr 12 '11)@cswiercz: This is basically how you have to do things in Mathematica - you then check results for a range of N and hope that that's enough. I was sure that sympy used to have an abstract matrix module... but I couldn't find it when I looked just now. Simon (Apr 12 '11) You say it's not the quickest way to do this (*this* being define lots of symbolic variables and populate a matrix with them), so what is the quickest way? benjaminfjones (Apr 19 '11) see 4 more comments