X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fmatrix_algebra.py;h=84aa8d28bb3de9e6797b9b0a209eb96c3a14dc22;hb=43783fbb6e8292a67506f6df876ab1de6dab68b1;hp=4049ef653a8a2688c2b9d4399f2b5386212aae47;hpb=ab0536f4db17eb78f3623927653b8f7f1a7e6808;p=sage.d.git diff --git a/mjo/matrix_algebra.py b/mjo/matrix_algebra.py index 4049ef6..84aa8d2 100644 --- a/mjo/matrix_algebra.py +++ b/mjo/matrix_algebra.py @@ -18,7 +18,7 @@ class MatrixAlgebraElement(IndexedFreeModuleElement): EXAMPLES:: - sage: M = MatrixAlgebra(QQbar,RDF,2) + sage: M = MatrixAlgebra(2, QQbar,RDF) sage: A = M.monomial((0,0,1)) + 4*M.monomial((0,1,1)) sage: A +-----+-----+ @@ -37,7 +37,7 @@ class MatrixAlgebraElement(IndexedFreeModuleElement): l[i][j] += v*e return l - def __repr__(self): + def _repr_(self): r""" Display this matrix as a table. @@ -50,7 +50,7 @@ class MatrixAlgebraElement(IndexedFreeModuleElement): EXAMPLES:: - sage: MatrixAlgebra(ZZ,ZZ,2).zero() + sage: MatrixAlgebra(2,ZZ,ZZ).zero() +---+---+ | 0 | 0 | +---+---+ @@ -71,7 +71,7 @@ class MatrixAlgebraElement(IndexedFreeModuleElement): EXAMPLES:: - sage: A = MatrixAlgebra(ZZ,ZZ,2) + sage: A = MatrixAlgebra(2,ZZ,ZZ) sage: A([[1,2],[3,4]]).list() [1, 2, 3, 4] @@ -88,7 +88,7 @@ class MatrixAlgebraElement(IndexedFreeModuleElement): EXAMPLES:: - sage: M = MatrixAlgebra(ZZ,ZZ,2)([[1,2],[3,4]]) + sage: M = MatrixAlgebra(2,ZZ,ZZ)([[1,2],[3,4]]) sage: M[0,0] 1 sage: M[0,1] @@ -117,7 +117,7 @@ class MatrixAlgebraElement(IndexedFreeModuleElement): sage: entries = MatrixSpace(ZZ,2) sage: scalars = ZZ - sage: M = MatrixAlgebra(entries, scalars, 2) + sage: M = MatrixAlgebra(2, entries, scalars) sage: I = entries.one() sage: Z = entries.zero() sage: M([[I,Z],[Z,I]]).trace() @@ -139,7 +139,7 @@ class MatrixAlgebraElement(IndexedFreeModuleElement): sage: set_random_seed() sage: entries = QuaternionAlgebra(QQ,-1,-1) - sage: M = MatrixAlgebra(entries, QQ, 3) + sage: M = MatrixAlgebra(3, entries, QQ) sage: M.random_element().matrix_space() == M True @@ -158,16 +158,37 @@ class MatrixAlgebra(CombinatorialFreeModule): the entries come from a commutative and associative ring. This is problematic in several interesting matrix algebras, like those where the entries are quaternions or octonions. + + SETUP:: + + sage: from mjo.matrix_algebra import MatrixAlgebra + + EXAMPLES:: + + The existence of a unit element is determined dynamically:: + + sage: MatrixAlgebra(2,ZZ,ZZ).one() + +---+---+ + | 1 | 0 | + +---+---+ + | 0 | 1 | + +---+---+ + """ Element = MatrixAlgebraElement - def __init__(self, entry_algebra, scalars, n, prefix="A", **kwargs): + def __init__(self, n, entry_algebra, scalars, prefix="A", **kwargs): category = MagmaticAlgebras(scalars).FiniteDimensional() category = category.WithBasis() if "Unital" in entry_algebra.category().axioms(): category = category.Unital() + entry_one = entry_algebra.one() + self.one = lambda: sum( (self.monomial((i,i,entry_one)) + for i in range(self.nrows()) ), + self.zero() ) + if "Associative" in entry_algebra.category().axioms(): category = category.Associative() @@ -177,14 +198,16 @@ class MatrixAlgebra(CombinatorialFreeModule): # sticking a "1" in each position doesn't give us a basis for # the space. We actually need to stick each of e0, e1, ... (a # basis for the entry algebra itself) into each position. - I = range(n) - J = range(n) self._entry_algebra = entry_algebra - entry_basis = entry_algebra.gens() - basis_indices = [(i,j,e) for i in range(n) - for j in range(n) - for e in entry_algebra.gens()] + # Needs to make the (overridden) method call when, for example, + # the entry algebra is the complex numbers and its gens() method + # lies to us. + entry_basis = self.entry_algebra_gens() + + basis_indices = [(i,j,e) for j in range(n) + for i in range(n) + for e in entry_basis] super().__init__(scalars, basis_indices, @@ -206,15 +229,54 @@ class MatrixAlgebra(CombinatorialFreeModule): """ return self._entry_algebra + def entry_algebra_gens(self): + r""" + Return a tuple of the generators of (that is, a basis for) the + entries of this matrix algebra. + + This can be overridden in subclasses to work around the + inconsistency in the ``gens()`` methods of the various + entry algebras. + """ + return self.entry_algebra().gens() + def nrows(self): return self._nrows ncols = nrows def product_on_basis(self, mon1, mon2): + r""" + + SETUP:: + + sage: from mjo.hurwitz import Octonions + sage: from mjo.matrix_algebra import MatrixAlgebra + + TESTS:: + + sage: O = Octonions(QQ) + sage: e = O.gens() + sage: e[2]*e[1] + -e3 + sage: A = MatrixAlgebra(2,O,QQ) + sage: A.product_on_basis( (0,0,e[2]), (0,0,e[1]) ) + +-----+---+ + | -e3 | 0 | + +-----+---+ + | 0 | 0 | + +-----+---+ + + """ (i,j,e1) = mon1 (k,l,e2) = mon2 if j == k: - return self.monomial((i,l,e1*e2)) + # If e1*e2 has a negative sign in front of it, + # then (i,l,e1*e2) won't be a monomial! + p = e1*e2 + if (i,l,p) in self.indices(): + return self.monomial((i,l,p)) + else: + return -self.monomial((i,l,-p)) else: return self.zero() @@ -229,7 +291,7 @@ class MatrixAlgebra(CombinatorialFreeModule): EXAMPLES:: - sage: A = MatrixAlgebra(QQbar, ZZ, 2) + sage: A = MatrixAlgebra(2, QQbar, ZZ) sage: A.from_list([[0,I],[-I,0]]) +----+---+ | 0 | I | @@ -273,42 +335,3 @@ class MatrixAlgebra(CombinatorialFreeModule): return self else: return self.from_list(elt) - - -class HurwitzMatrixAlgebraElement(MatrixAlgebraElement): - def is_hermitian(self): - r""" - - SETUP:: - - sage: from mjo.matrix_algebra import HurwitzMatrixAlgebra - - EXAMPLES:: - - sage: A = HurwitzMatrixAlgebra(QQbar, ZZ, 2) - sage: M = A([ [ 0,I], - ....: [-I,0] ]) - sage: M.is_hermitian() - True - - """ - return all( self[i,j] == self[j,i].conjugate() - for i in range(self.nrows()) - for j in range(self.ncols()) ) - - -class HurwitzMatrixAlgebra(MatrixAlgebra): - Element = HurwitzMatrixAlgebraElement - - def one(self): - r""" - SETUP:: - - sage: from mjo.matrix_algebra import HurwitzMatrixAlgebra - - """ - return sum( (self.monomial((i,i,self.entry_algebra().one())) - for i in range(self.nrows()) ), - self.zero() ) - -