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.algebras.finite_dimensional_algebras.finite_dimensional_algebra import FiniteDimensionalAlgebra
   9 from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra_element import FiniteDimensionalAlgebraElement
  10
  11 class FiniteDimensionalEuclideanJordanAlgebra(FiniteDimensionalAlgebra):
  12     @staticmethod
  13     def __classcall__(cls, field, mult_table, names='e', category=None):
  14         fda = super(FiniteDimensionalEuclideanJordanAlgebra, cls)
  15         return fda.__classcall_private__(cls,
  16                                          field,
  17                                          mult_table,
  18                                          names,
  19                                          category)
  20
  21     def __init__(self, field, mult_table, names='e', category=None):
  22         fda = super(FiniteDimensionalEuclideanJordanAlgebra, self)
  23         fda.__init__(field, mult_table, names, category)
  24
  25
  26     def _repr_(self):
  27         """
  28         Return a string representation of ``self``.
  29         """
  30         return "Euclidean Jordan algebra of degree {} over {}".format(self.degree(), self.base_ring())
  31
  32     def rank(self):
  33         """
  34         Return the rank of this EJA.
  35         """
  36         raise NotImplementedError
  37
  38
  39     class Element(FiniteDimensionalAlgebraElement):
  40         """
  41         An element of a Euclidean Jordan algebra.
  42
  43         Since EJAs are commutative, the "right multiplication" matrix is
  44         also the left multiplication matrix and must be symmetric::
  45
  46             sage: set_random_seed()
  47             sage: J = eja_ln(5)
  48             sage: J.random_element().matrix().is_symmetric()
  49             True
  50
  51         """
  52
  53         def __pow__(self, n):
  54             """
  55             Return ``self`` raised to the power ``n``.
  56
  57             Jordan algebras are always power-associative; see for
  58             example Faraut and Koranyi, Proposition II.1.2 (ii).
  59             """
  60             A = self.parent()
  61             if n == 0:
  62                 return A.one()
  63             elif n == 1:
  64                 return self
  65             else:
  66                 return A.element_class(A, self.vector()*(self.matrix()**(n-1)))
  67
  68
  69         def span_of_powers(self):
  70             """
  71             Return the vector space spanned by successive powers of
  72             this element.
  73             """
  74             # The dimension of the subalgebra can't be greater than
  75             # the big algebra, so just put everything into a list
  76             # and let span() get rid of the excess.
  77             V = self.vector().parent()
  78             return V.span( (self**d).vector() for d in xrange(V.dimension()) )
  79
  80
  81         def degree(self):
  82             """
  83             Compute the degree of this element the straightforward way
  84             according to the definition; by appending powers to a list
  85             and figuring out its dimension (that is, whether or not
  86             they're linearly dependent).
  87
  88             EXAMPLES::
  89
  90                 sage: J = eja_ln(4)
  91                 sage: J.one().degree()
  92                 1
  93                 sage: e0,e1,e2,e3 = J.gens()
  94                 sage: (e0 - e1).degree()
  95                 2
  96
  97             In the spin factor algebra (of rank two), all elements that
  98             aren't multiples of the identity are regular::
  99
 100                 sage: set_random_seed()
 101                 sage: n = ZZ.random_element(1,10).abs()
 102                 sage: J = eja_ln(n)
 103                 sage: x = J.random_element()
 104                 sage: x == x.coefficient(0)*J.one() or x.degree() == 2
 105                 True
 106
 107             """
 108             return self.span_of_powers().dimension()
 109
 110
 111         def minimal_polynomial(self):
 112             return self.matrix().minimal_polynomial()
 113
 114         def characteristic_polynomial(self):
 115             return self.matrix().characteristic_polynomial()
 116
 117
 118 def eja_rn(dimension, field=QQ):
 119     """
 120     Return the Euclidean Jordan Algebra corresponding to the set
 121     `R^n` under the Hadamard product.
 122
 123     EXAMPLES:
 124
 125     This multiplication table can be verified by hand::
 126
 127         sage: J = eja_rn(3)
 128         sage: e0,e1,e2 = J.gens()
 129         sage: e0*e0
 130         e0
 131         sage: e0*e1
 132         0
 133         sage: e0*e2
 134         0
 135         sage: e1*e1
 136         e1
 137         sage: e1*e2
 138         0
 139         sage: e2*e2
 140         e2
 141
 142     """
 143     # The FiniteDimensionalAlgebra constructor takes a list of
 144     # matrices, the ith representing right multiplication by the ith
 145     # basis element in the vector space. So if e_1 = (1,0,0), then
 146     # right (Hadamard) multiplication of x by e_1 picks out the first
 147     # component of x; and likewise for the ith basis element e_i.
 148     Qs = [ matrix(field, dimension, dimension, lambda k,j: 1*(k == j == i))
 149            for i in xrange(dimension) ]
 150
 151     return FiniteDimensionalEuclideanJordanAlgebra(field,Qs)
 152
 153
 154 def eja_ln(dimension, field=QQ):
 155     """
 156     Return the Jordan algebra corresponding to the Lorentz "ice cream"
 157     cone of the given ``dimension``.
 158
 159     EXAMPLES:
 160
 161     This multiplication table can be verified by hand::
 162
 163         sage: J = eja_ln(4)
 164         sage: e0,e1,e2,e3 = J.gens()
 165         sage: e0*e0
 166         e0
 167         sage: e0*e1
 168         e1
 169         sage: e0*e2
 170         e2
 171         sage: e0*e3
 172         e3
 173         sage: e1*e2
 174         0
 175         sage: e1*e3
 176         0
 177         sage: e2*e3
 178         0
 179
 180     In one dimension, this is the reals under multiplication::
 181
 182       sage: J1 = eja_ln(1)
 183       sage: J2 = eja_rn(1)
 184       sage: J1 == J2
 185       True
 186
 187     """
 188     Qs = []
 189     id_matrix = identity_matrix(field,dimension)
 190     for i in xrange(dimension):
 191         ei = id_matrix.column(i)
 192         Qi = zero_matrix(field,dimension)
 193         Qi.set_row(0, ei)
 194         Qi.set_column(0, ei)
 195         Qi += diagonal_matrix(dimension, [ei[0]]*dimension)
 196         # The addition of the diagonal matrix adds an extra ei[0] in the
 197         # upper-left corner of the matrix.
 198         Qi[0,0] = Qi[0,0] * ~field(2)
 199         Qs.append(Qi)
 200
 201     return FiniteDimensionalEuclideanJordanAlgebra(field,Qs)