X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fmatrix_algebra.py;h=8491f277d5cc81b955c6ddaed0298549d15aa779;hb=f721f283cea741c54e4a2b26d63094dd6c396c6a;hp=a67a9b4a1c0692d31aceffa3097343aa0df99dae;hpb=e8d6e417a08d9f82398dd2aec6a28ced656fab53;p=sage.d.git diff --git a/mjo/matrix_algebra.py b/mjo/matrix_algebra.py index a67a9b4..8491f27 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 +-----+-----+ @@ -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 @@ -167,7 +167,7 @@ class MatrixAlgebra(CombinatorialFreeModule): The existence of a unit element is determined dynamically:: - sage: MatrixAlgebra(ZZ,ZZ,2).one() + sage: MatrixAlgebra(2,ZZ,ZZ).one() +---+---+ | 1 | 0 | +---+---+ @@ -177,7 +177,7 @@ class MatrixAlgebra(CombinatorialFreeModule): """ 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() @@ -198,14 +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, @@ -227,6 +229,92 @@ 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 _entry_algebra_element_to_vector(self, entry): + r""" + Return a vector representation (of length equal to the cardinality + of :meth:`entry_algebra_gens`) of the given ``entry``. + + This can be overridden in subclasses to work around the fact that + real numbers, complex numbers, quaternions, et cetera, all require + different incantations to turn them into a vector. + + It only makes sense to "guess" here in the superclass when no + subclass that overrides :meth:`entry_algebra_gens` exists. So + if you have a special subclass for your annoying entry algebra, + override this with the correct implementation there instead of + adding a bunch of awkward cases to this superclass method. + + SETUP:: + + sage: from mjo.hurwitz import Octonions + sage: from mjo.matrix_algebra import MatrixAlgebra + + EXAMPLES: + + Real numbers:: + + sage: A = MatrixAlgebra(1, AA, QQ) + sage: A._entry_algebra_element_to_vector(AA(17)) + (17) + + Octonions:: + + sage: A = MatrixAlgebra(1, Octonions(), QQ) + sage: e = A.entry_algebra_gens() + sage: A._entry_algebra_element_to_vector(e[0]) + (1, 0, 0, 0, 0, 0, 0, 0) + sage: A._entry_algebra_element_to_vector(e[1]) + (0, 1, 0, 0, 0, 0, 0, 0) + sage: A._entry_algebra_element_to_vector(e[2]) + (0, 0, 1, 0, 0, 0, 0, 0) + sage: A._entry_algebra_element_to_vector(e[3]) + (0, 0, 0, 1, 0, 0, 0, 0) + sage: A._entry_algebra_element_to_vector(e[4]) + (0, 0, 0, 0, 1, 0, 0, 0) + sage: A._entry_algebra_element_to_vector(e[5]) + (0, 0, 0, 0, 0, 1, 0, 0) + sage: A._entry_algebra_element_to_vector(e[6]) + (0, 0, 0, 0, 0, 0, 1, 0) + sage: A._entry_algebra_element_to_vector(e[7]) + (0, 0, 0, 0, 0, 0, 0, 1) + + Sage matrices:: + + sage: MS = MatrixSpace(QQ,2) + sage: A = MatrixAlgebra(1, MS, QQ) + sage: A._entry_algebra_element_to_vector(MS([[1,2],[3,4]])) + (1, 2, 3, 4) + + """ + if hasattr(entry, 'to_vector'): + return entry.to_vector() + + from sage.modules.free_module import VectorSpace + d = len(self.entry_algebra_gens()) + V = VectorSpace(self.entry_algebra().base_ring(), d) + + if hasattr(entry, 'list'): + # sage matrices + return V(entry.list()) + + # This works in AA, and will crash if it doesn't know what to + # do, and that's fine because then I don't know what to do + # either. + return V((entry,)) + + + def nrows(self): return self._nrows ncols = nrows @@ -236,7 +324,7 @@ class MatrixAlgebra(CombinatorialFreeModule): SETUP:: - sage: from mjo.octonions import Octonions + sage: from mjo.hurwitz import Octonions sage: from mjo.matrix_algebra import MatrixAlgebra TESTS:: @@ -245,7 +333,7 @@ class MatrixAlgebra(CombinatorialFreeModule): sage: e = O.gens() sage: e[2]*e[1] -e3 - sage: A = MatrixAlgebra(O,QQ,2) + sage: A = MatrixAlgebra(2,O,QQ) sage: A.product_on_basis( (0,0,e[2]), (0,0,e[1]) ) +-----+---+ | -e3 | 0 | @@ -278,7 +366,7 @@ 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 | @@ -322,29 +410,3 @@ 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