]>
gitweb.michael.orlitzky.com - sage.d.git/blob - mjo/matrix_algebra.py
d67347b3aa1252acaf165448d1ea35ebb4a77cd7
1 from sage
.misc
.table
import table
2 from sage
.categories
.magmatic_algebras
import MagmaticAlgebras
3 from sage
.misc
.cachefunc
import cached_method
4 from sage
.combinat
.free_module
import CombinatorialFreeModule
5 from sage
.modules
.with_basis
.indexed_element
import IndexedFreeModuleElement
7 class MatrixAlgebraElement(IndexedFreeModuleElement
):
9 return self
.parent().nrows()
17 sage: from mjo.matrix_algebra import MatrixAlgebra
21 sage: M = MatrixAlgebra(2, QQbar,RDF)
22 sage: A = M.monomial((0,0,1)) + 4*M.monomial((0,1,1))
33 zero
= self
.parent().entry_algebra().zero()
34 l
= [[zero
for j
in range(self
.ncols())] for i
in range(self
.nrows())]
35 for (k
,v
) in self
.monomial_coefficients().items():
42 Display this matrix as a table.
44 The SageMath Matrix class representation is not easily reusable,
45 but using a table fakes it.
49 sage: from mjo.matrix_algebra import MatrixAlgebra
53 sage: MatrixAlgebra(2,ZZ,ZZ).zero()
61 return table(self
.rows(), frame
=True)._repr
_()
66 Return one long list of this matrix's entries.
70 sage: from mjo.matrix_algebra import MatrixAlgebra
74 sage: A = MatrixAlgebra(2,ZZ,ZZ)
75 sage: A([[1,2],[3,4]]).list()
79 return sum( self
.rows(), [] )
82 def __getitem__(self
, indices
):
87 sage: from mjo.matrix_algebra import MatrixAlgebra
91 sage: M = MatrixAlgebra(2,ZZ,ZZ)([[1,2],[3,4]])
103 return self
.rows()[i
][j
]
107 Return the sum of this matrix's diagonal entries.
111 sage: from mjo.matrix_algebra import MatrixAlgebra
115 The trace (being a sum of entries) belongs to the same algebra
116 as those entries, and NOT the scalar ring::
118 sage: entries = MatrixSpace(ZZ,2)
120 sage: M = MatrixAlgebra(2, entries, scalars)
121 sage: I = entries.one()
122 sage: Z = entries.zero()
123 sage: M([[I,Z],[Z,I]]).trace()
128 zero
= self
.parent().entry_algebra().zero()
129 return sum( (self
[i
,i
] for i
in range(self
.nrows())), zero
)
131 def matrix_space(self
):
136 sage: from mjo.matrix_algebra import MatrixAlgebra
140 sage: set_random_seed()
141 sage: entries = QuaternionAlgebra(QQ,-1,-1)
142 sage: M = MatrixAlgebra(3, entries, QQ)
143 sage: M.random_element().matrix_space() == M
150 class MatrixAlgebra(CombinatorialFreeModule
):
152 An algebra of ``n``-by-``n`` matrices over an arbitrary scalar
153 ring whose entries come from a magmatic algebra that need not
154 be the same as the scalars.
156 The usual matrix spaces in SageMath don't support separate spaces
157 for the entries and the scalars; in particular they assume that
158 the entries come from a commutative and associative ring. This
159 is problematic in several interesting matrix algebras, like those
160 where the entries are quaternions or octonions.
164 sage: from mjo.matrix_algebra import MatrixAlgebra
168 The existence of a unit element is determined dynamically::
170 sage: MatrixAlgebra(2,ZZ,ZZ).one()
178 Element
= MatrixAlgebraElement
180 def __init__(self
, n
, entry_algebra
, scalars
, prefix
="A", **kwargs
):
182 category
= MagmaticAlgebras(scalars
).FiniteDimensional()
183 category
= category
.WithBasis()
185 if "Unital" in entry_algebra
.category().axioms():
186 category
= category
.Unital()
187 entry_one
= entry_algebra
.one()
188 self
.one
= lambda: sum( (self
.monomial((i
,i
,entry_one
))
189 for i
in range(self
.nrows()) ),
192 if "Associative" in entry_algebra
.category().axioms():
193 category
= category
.Associative()
197 # Since the scalar ring is real but the entries are not,
198 # sticking a "1" in each position doesn't give us a basis for
199 # the space. We actually need to stick each of e0, e1, ... (a
200 # basis for the entry algebra itself) into each position.
203 self
._entry
_algebra
= entry_algebra
205 # Needs to make the (overridden) method call when, for example,
206 # the entry algebra is the complex numbers and its gens() method
208 entry_basis
= self
.entry_algebra_gens()
210 basis_indices
= [(i
,j
,e
) for i
in range(n
)
212 for e
in entry_basis
]
214 super().__init
__(scalars
,
221 return ("Module of %d by %d matrices with entries in %s"
222 " over the scalar ring %s" %
225 self
.entry_algebra(),
228 def entry_algebra(self
):
230 Return the algebra that our elements' entries come from.
232 return self
._entry
_algebra
234 def entry_algebra_gens(self
):
236 Return a tuple of the generators of (that is, a basis for) the
237 entries of this matrix algebra.
239 This can be overridden in subclasses to work around the
240 inconsistency in the ``gens()`` methods of the various
243 return self
.entry_algebra().gens()
249 def product_on_basis(self
, mon1
, mon2
):
254 sage: from mjo.hurwitz import Octonions
255 sage: from mjo.matrix_algebra import MatrixAlgebra
259 sage: O = Octonions(QQ)
263 sage: A = MatrixAlgebra(2,O,QQ)
264 sage: A.product_on_basis( (0,0,e[2]), (0,0,e[1]) )
275 # If e1*e2 has a negative sign in front of it,
276 # then (i,l,e1*e2) won't be a monomial!
278 if (i
,l
,p
) in self
.indices():
279 return self
.monomial((i
,l
,p
))
281 return -self
.monomial((i
,l
,-p
))
285 def from_list(self
, entries
):
287 Construct an element of this algebra from a list of lists of
292 sage: from mjo.matrix_algebra import MatrixAlgebra
296 sage: A = MatrixAlgebra(2, QQbar, ZZ)
297 sage: A.from_list([[0,I],[-I,0]])
308 ncols
= len(entries
[0])
310 if (not all( len(r
) == ncols
for r
in entries
)) or (ncols
!= nrows
):
311 raise ValueError("list must be square")
314 if e_ij
in self
.entry_algebra():
315 # Don't re-create an element if it already lives where
320 # This branch works with e.g. QQbar, where no
321 # to/from_vector() methods are available.
322 return self
.entry_algebra()(e_ij
)
324 # We have to pass through vectors to convert from the
325 # given entry algebra to ours. Otherwise we can fail to
326 # convert an element of (for example) Octonions(QQ) to
328 return self
.entry_algebra().from_vector(e_ij
.to_vector())
330 return sum( (self
.monomial( (i
,j
, convert(entries
[i
][j
])) )
331 for i
in range(nrows
)
332 for j
in range(ncols
) ),
335 def _element_constructor_(self
, elt
):
339 return self
.from_list(elt
)