mjo/eja/euclidean_jordan_algebra.py

   1 """
   2 Euclidean Jordan Algebras. These are formally-real Jordan Algebras;
   3 specifically those where u^2 + v^2 = 0 implies that u = v = 0. They
   4 are used in optimization, and have some additional nice methods beyond
   5 what can be supported in a general Jordan Algebra.
   6 """
   7
   8 from sage.categories.magmatic_algebras import MagmaticAlgebras
   9 from sage.structure.element import is_Matrix
  10 from sage.structure.category_object import normalize_names
  11
  12 from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra import FiniteDimensionalAlgebra
  13 from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_element import FiniteDimensionalAlgebraElement
  14
  15 class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
  16     @staticmethod
  17     def __classcall_private__(cls,
  18                               field,
  19                               mult_table,
  20                               names='e',
  21                               assume_associative=False,
  22                               category=None,
  23                               rank=None):
  24         n = len(mult_table)
  25         mult_table = [b.base_extend(field) for b in mult_table]
  26         for b in mult_table:
  27             b.set_immutable()
  28             if not (is_Matrix(b) and b.dimensions() == (n, n)):
  29                 raise ValueError("input is not a multiplication table")
  30         mult_table = tuple(mult_table)
  31
  32         cat = MagmaticAlgebras(field).FiniteDimensional().WithBasis()
  33         cat.or_subcategory(category)
  34         if assume_associative:
  35             cat = cat.Associative()
  36
  37         names = normalize_names(n, names)
  38
  39         fda = super(FiniteDimensionalEuclideanJordanAlgebra, cls)
  40         return fda.__classcall__(cls,
  41                                  field,
  42                                  mult_table,
  43                                  assume_associative=assume_associative,
  44                                  names=names,
  45                                  category=cat,
  46                                  rank=rank)
  47
  48
  49     def __init__(self, field,
  50                  mult_table,
  51                  names='e',
  52                  assume_associative=False,
  53                  category=None,
  54                  rank=None):
  55         self._rank = rank
  56         fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
  57         fda.__init__(field,
  58                      mult_table,
  59                      names=names,
  60                      category=category)
  61
  62
  63     def _repr_(self):
  64         """
  65         Return a string representation of ``self``.
  66         """
  67         fmt = "Euclidean Jordan algebra of degree {} over {}"
  68         return fmt.format(self.degree(), self.base_ring())
  69
  70     def rank(self):
  71         """
  72         Return the rank of this EJA.
  73         """
  74         if self._rank is None:
  75             raise ValueError("no rank specified at genesis")
  76         else:
  77             return self._rank
  78
  79
  80     class Element(FiniteDimensionalAlgebraElement):
  81         """
  82         An element of a Euclidean Jordan algebra.
  83
  84         Since EJAs are commutative, the "right multiplication" matrix is
  85         also the left multiplication matrix and must be symmetric::
  86
  87             sage: set_random_seed()
  88             sage: n = ZZ.random_element(1,10).abs()
  89             sage: J = eja_rn(5)
  90             sage: J.random_element().matrix().is_symmetric()
  91             True
  92             sage: J = eja_ln(5)
  93             sage: J.random_element().matrix().is_symmetric()
  94             True
  95
  96         """
  97
  98         def __pow__(self, n):
  99             """
 100             Return ``self`` raised to the power ``n``.
 101
 102             Jordan algebras are always power-associative; see for
 103             example Faraut and Koranyi, Proposition II.1.2 (ii).
 104             """
 105             A = self.parent()
 106             if n == 0:
 107                 return A.one()
 108             elif n == 1:
 109                 return self
 110             else:
 111                 return A.element_class(A, self.vector()*(self.matrix()**(n-1)))
 112
 113
 114         def span_of_powers(self):
 115             """
 116             Return the vector space spanned by successive powers of
 117             this element.
 118             """
 119             # The dimension of the subalgebra can't be greater than
 120             # the big algebra, so just put everything into a list
 121             # and let span() get rid of the excess.
 122             V = self.vector().parent()
 123             return V.span( (self**d).vector() for d in xrange(V.dimension()) )
 124
 125
 126         def degree(self):
 127             """
 128             Compute the degree of this element the straightforward way
 129             according to the definition; by appending powers to a list
 130             and figuring out its dimension (that is, whether or not
 131             they're linearly dependent).
 132
 133             EXAMPLES::
 134
 135                 sage: J = eja_ln(4)
 136                 sage: J.one().degree()
 137                 1
 138                 sage: e0,e1,e2,e3 = J.gens()
 139                 sage: (e0 - e1).degree()
 140                 2
 141
 142             In the spin factor algebra (of rank two), all elements that
 143             aren't multiples of the identity are regular::
 144
 145                 sage: set_random_seed()
 146                 sage: n = ZZ.random_element(1,10).abs()
 147                 sage: J = eja_ln(n)
 148                 sage: x = J.random_element()
 149                 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
 150                 True
 151
 152             """
 153             return self.span_of_powers().dimension()
 154
 155
 156         def subalgebra_generated_by(self):
 157             """
 158             Return the associative subalgebra of the parent EJA generated
 159             by this element.
 160
 161             TESTS::
 162
 163                 sage: set_random_seed()
 164                 sage: n = ZZ.random_element(1,10).abs()
 165                 sage: J = eja_rn(n)
 166                 sage: x = J.random_element()
 167                 sage: x.subalgebra_generated_by().is_associative()
 168                 True
 169                 sage: J = eja_ln(n)
 170                 sage: x = J.random_element()
 171                 sage: x.subalgebra_generated_by().is_associative()
 172                 True
 173
 174             """
 175             # First get the subspace spanned by the powers of myself...
 176             V = self.span_of_powers()
 177             F = self.base_ring()
 178
 179             # Now figure out the entries of the right-multiplication
 180             # matrix for the successive basis elements b0, b1,... of
 181             # that subspace.
 182             mats = []
 183             for b_right in V.basis():
 184                 eja_b_right = self.parent()(b_right)
 185                 b_right_rows = []
 186                 # The first row of the right-multiplication matrix by
 187                 # b1 is what we get if we apply that matrix to b1. The
 188                 # second row of the right multiplication matrix by b1
 189                 # is what we get when we apply that matrix to b2...
 190                 for b_left in V.basis():
 191                     eja_b_left = self.parent()(b_left)
 192                     # Multiply in the original EJA, but then get the
 193                     # coordinates from the subalgebra in terms of its
 194                     # basis.
 195                     this_row = V.coordinates((eja_b_left*eja_b_right).vector())
 196                     b_right_rows.append(this_row)
 197                 b_right_matrix = matrix(F, b_right_rows)
 198                 mats.append(b_right_matrix)
 199
 200             # It's an algebra of polynomials in one element, and EJAs
 201             # are power-associative.
 202             return FiniteDimensionalEuclideanJordanAlgebra(F, mats, assume_associative=True)
 203
 204
 205         def minimal_polynomial(self):
 206             """
 207             EXAMPLES::
 208
 209                 sage: set_random_seed()
 210                 sage: n = ZZ.random_element(1,10).abs()
 211                 sage: J = eja_rn(n)
 212                 sage: x = J.random_element()
 213                 sage: x.degree() == x.minimal_polynomial().degree()
 214                 True
 215
 216             ::
 217
 218                 sage: set_random_seed()
 219                 sage: n = ZZ.random_element(1,10).abs()
 220                 sage: J = eja_ln(n)
 221                 sage: x = J.random_element()
 222                 sage: x.degree() == x.minimal_polynomial().degree()
 223                 True
 224
 225             The minimal polynomial and the characteristic polynomial coincide
 226             and are known (see Alizadeh, Example 11.11) for all elements of
 227             the spin factor algebra that aren't scalar multiples of the
 228             identity::
 229
 230                 sage: set_random_seed()
 231                 sage: n = ZZ.random_element(2,10).abs()
 232                 sage: J = eja_ln(n)
 233                 sage: y = J.random_element()
 234                 sage: while y == y.coefficient(0)*J.one():
 235                 ....:     y = J.random_element()
 236                 sage: y0 = y.vector()[0]
 237                 sage: y_bar = y.vector()[1:]
 238                 sage: actual = y.minimal_polynomial()
 239                 sage: x = SR.symbol('x', domain='real')
 240                 sage: expected = x^2 - 2*y0*x + (y0^2 - norm(y_bar)^2)
 241                 sage: bool(actual == expected)
 242                 True
 243
 244             """
 245             # The element we're going to call "minimal_polynomial()" on.
 246             # Either myself, interpreted as an element of a finite-
 247             # dimensional algebra, or an element of an associative
 248             # subalgebra.
 249             elt = None
 250
 251             if self.parent().is_associative():
 252                 elt = FiniteDimensionalAlgebraElement(self.parent(), self)
 253             else:
 254                 V = self.span_of_powers()
 255                 assoc_subalg = self.subalgebra_generated_by()
 256                 # Mis-design warning: the basis used for span_of_powers()
 257                 # and subalgebra_generated_by() must be the same, and in
 258                 # the same order!
 259                 elt = assoc_subalg(V.coordinates(self.vector()))
 260
 261             # Recursive call, but should work since elt lives in an
 262             # associative algebra.
 263             return elt.minimal_polynomial()
 264
 265
 266         def is_nilpotent(self):
 267             """
 268             Return whether or not some power of this element is zero.
 269
 270             The superclass method won't work unless we're in an
 271             associative algebra, and we aren't. However, we generate
 272             an assocoative subalgebra and we're nilpotent there if and
 273             only if we're nilpotent here (probably).
 274
 275             TESTS:
 276
 277             The identity element is never nilpotent::
 278
 279                 sage: set_random_seed()
 280                 sage: n = ZZ.random_element(2,10).abs()
 281                 sage: J = eja_rn(n)
 282                 sage: J.one().is_nilpotent()
 283                 False
 284                 sage: J = eja_ln(n)
 285                 sage: J.one().is_nilpotent()
 286                 False
 287
 288             The additive identity is always nilpotent::
 289
 290                 sage: set_random_seed()
 291                 sage: n = ZZ.random_element(2,10).abs()
 292                 sage: J = eja_rn(n)
 293                 sage: J.zero().is_nilpotent()
 294                 True
 295                 sage: J = eja_ln(n)
 296                 sage: J.zero().is_nilpotent()
 297                 True
 298
 299             """
 300             # The element we're going to call "is_nilpotent()" on.
 301             # Either myself, interpreted as an element of a finite-
 302             # dimensional algebra, or an element of an associative
 303             # subalgebra.
 304             elt = None
 305
 306             if self.parent().is_associative():
 307                 elt = FiniteDimensionalAlgebraElement(self.parent(), self)
 308             else:
 309                 V = self.span_of_powers()
 310                 assoc_subalg = self.subalgebra_generated_by()
 311                 # Mis-design warning: the basis used for span_of_powers()
 312                 # and subalgebra_generated_by() must be the same, and in
 313                 # the same order!
 314                 elt = assoc_subalg(V.coordinates(self.vector()))
 315
 316             # Recursive call, but should work since elt lives in an
 317             # associative algebra.
 318             return elt.is_nilpotent()
 319
 320
 321         def characteristic_polynomial(self):
 322             return self.matrix().characteristic_polynomial()
 323
 324
 325 def eja_rn(dimension, field=QQ):
 326     """
 327     Return the Euclidean Jordan Algebra corresponding to the set
 328     `R^n` under the Hadamard product.
 329
 330     EXAMPLES:
 331
 332     This multiplication table can be verified by hand::
 333
 334         sage: J = eja_rn(3)
 335         sage: e0,e1,e2 = J.gens()
 336         sage: e0*e0
 337         e0
 338         sage: e0*e1
 339         0
 340         sage: e0*e2
 341         0
 342         sage: e1*e1
 343         e1
 344         sage: e1*e2
 345         0
 346         sage: e2*e2
 347         e2
 348
 349     """
 350     # The FiniteDimensionalAlgebra constructor takes a list of
 351     # matrices, the ith representing right multiplication by the ith
 352     # basis element in the vector space. So if e_1 = (1,0,0), then
 353     # right (Hadamard) multiplication of x by e_1 picks out the first
 354     # component of x; and likewise for the ith basis element e_i.
 355     Qs = [ matrix(field, dimension, dimension, lambda k,j: 1*(k == j == i))
 356            for i in xrange(dimension) ]
 357
 358     return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=dimension)
 359
 360
 361 def eja_ln(dimension, field=QQ):
 362     """
 363     Return the Jordan algebra corresponding to the Lorentz "ice cream"
 364     cone of the given ``dimension``.
 365
 366     EXAMPLES:
 367
 368     This multiplication table can be verified by hand::
 369
 370         sage: J = eja_ln(4)
 371         sage: e0,e1,e2,e3 = J.gens()
 372         sage: e0*e0
 373         e0
 374         sage: e0*e1
 375         e1
 376         sage: e0*e2
 377         e2
 378         sage: e0*e3
 379         e3
 380         sage: e1*e2
 381         0
 382         sage: e1*e3
 383         0
 384         sage: e2*e3
 385         0
 386
 387     In one dimension, this is the reals under multiplication::
 388
 389       sage: J1 = eja_ln(1)
 390       sage: J2 = eja_rn(1)
 391       sage: J1 == J2
 392       True
 393
 394     """
 395     Qs = []
 396     id_matrix = identity_matrix(field,dimension)
 397     for i in xrange(dimension):
 398         ei = id_matrix.column(i)
 399         Qi = zero_matrix(field,dimension)
 400         Qi.set_row(0, ei)
 401         Qi.set_column(0, ei)
 402         Qi += diagonal_matrix(dimension, [ei[0]]*dimension)
 403         # The addition of the diagonal matrix adds an extra ei[0] in the
 404         # upper-left corner of the matrix.
 405         Qi[0,0] = Qi[0,0] * ~field(2)
 406         Qs.append(Qi)
 407
 408     # The rank of the spin factor algebra is two, UNLESS we're in a
 409     # one-dimensional ambient space (the rank is bounded by the
 410     # ambient dimension).
 411     rank = min(dimension,2)
 412     return FiniteDimensionalEuclideanJordanAlgebra(field,Qs,rank=rank)