l[i][j] += v*e
return l
- def __repr__(self):
+ def _repr_(self):
r"""
Display this matrix as a table.
EXAMPLES::
- sage: MatrixAlgebra(ZZ,ZZ,2).one()
+ sage: MatrixAlgebra(ZZ,ZZ,2).zero()
+---+---+
- | 1 | 0 |
+ | 0 | 0 |
+---+---+
- | 0 | 1 |
+ | 0 | 0 |
+---+---+
"""
EXAMPLES::
- sage: MatrixAlgebra(ZZ,ZZ,2).one().list()
- [1, 0, 0, 1]
+ sage: A = MatrixAlgebra(ZZ,ZZ,2)
+ sage: A([[1,2],[3,4]]).list()
+ [1, 2, 3, 4]
"""
return sum( self.rows(), [] )
EXAMPLES::
- sage: M = MatrixAlgebra(ZZ,ZZ,2).one()
+ sage: M = MatrixAlgebra(ZZ,ZZ,2)([[1,2],[3,4]])
sage: M[0,0]
1
sage: M[0,1]
- 0
+ 2
sage: M[1,0]
- 0
+ 3
sage: M[1,1]
- 1
+ 4
"""
i,j = indices
sage: entries = MatrixSpace(ZZ,2)
sage: scalars = ZZ
sage: M = MatrixAlgebra(entries, scalars, 2)
- sage: M.one().trace()
+ sage: I = entries.one()
+ sage: Z = entries.zero()
+ sage: M([[I,Z],[Z,I]]).trace()
[2 0]
[0 2]
"""
return self.parent()
- # onlt valid in HurwitzMatrixAlgebra subclass
- # def is_hermitian(self):
- # r"""
-
- # SETUP::
-
- # sage: from mjo.octonions import OctonionMatrixAlgebra
-
- # EXAMPLES::
-
- # sage: MS = OctonionMatrixAlgebra(3)
- # sage: MS.one().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 MatrixAlgebra(CombinatorialFreeModule):
r"""
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(ZZ,ZZ,2).one()
+ +---+---+
+ | 1 | 0 |
+ +---+---+
+ | 0 | 1 |
+ +---+---+
+
"""
Element = MatrixAlgebraElement
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()
ncols = nrows
def product_on_basis(self, mon1, mon2):
- (i,j,e1) = mon1
- (k,l,e2) = mon2
- if j == k:
- return self.monomial((i,l,e1*e2))
- else:
- return self.zero()
-
- # TODO: only makes sense if I'm unital.
- def one(self):
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(O,QQ,2)
+ sage: A.product_on_basis( (0,0,e[2]), (0,0,e[1]) )
+ +-----+---+
+ | -e3 | 0 |
+ +-----+---+
+ | 0 | 0 |
+ +-----+---+
+
"""
- return sum( (self.monomial((i,i,self.entry_algebra().one()))
- for i in range(self.nrows()) ),
- self.zero() )
+ (i,j,e1) = mon1
+ (k,l,e2) = mon2
+ if j == k:
+ # 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()
def from_list(self, entries):
r"""
sage: from mjo.matrix_algebra import MatrixAlgebra
+ EXAMPLES::
+
+ sage: A = MatrixAlgebra(QQbar, ZZ, 2)
+ sage: A.from_list([[0,I],[-I,0]])
+ +----+---+
+ | 0 | I |
+ +----+---+
+ | -I | 0 |
+ +----+---+
+
"""
nrows = len(entries)
ncols = 0
raise ValueError("list must be square")
def convert(e_ij):
- # We have to pass through vectors to convert from the
- # given entry algebra to ours. Otherwise we can fail
- # to convert an element of (for example) Octonions(QQ)
- # to Octonions(AA).
- return self.entry_algebra().from_vector(e_ij.to_vector())
+ if e_ij in self.entry_algebra():
+ # Don't re-create an element if it already lives where
+ # it should!
+ return e_ij
+
+ try:
+ # This branch works with e.g. QQbar, where no
+ # to/from_vector() methods are available.
+ return self.entry_algebra()(e_ij)
+ except TypeError:
+ # We have to pass through vectors to convert from the
+ # given entry algebra to ours. Otherwise we can fail to
+ # convert an element of (for example) Octonions(QQ) to
+ # Octonions(AA).
+ return self.entry_algebra().from_vector(e_ij.to_vector())
return sum( (self.monomial( (i,j, convert(entries[i][j])) )
for i in range(nrows)
for j in range(ncols) ),
self.zero() )
+
+ def _element_constructor_(self, elt):
+ if elt in self:
+ return self
+ else:
+ return self.from_list(elt)
projects / sage.d.git / blobdiff