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
  44         def __pow__(self, n):
  45             """
  46             Return ``self`` raised to the power ``n``.
  47
  48             Jordan algebras are always power-associative; see for
  49             example Faraut and Koranyi, Proposition II.1.2 (ii).
  50             """
  51             A = self.parent()
  52             if n == 0:
  53                 return A.one()
  54             elif n == 1:
  55                 return self
  56             else:
  57                 return A.element_class(A, self.vector()*(self.matrix()**(n-1)))
  58
  59
  60         def span_of_powers(self):
  61             """
  62             Return the vector space spanned by successive powers of
  63             this element.
  64             """
  65             # The dimension of the subalgebra can't be greater than
  66             # the big algebra, so just put everything into a list
  67             # and let span() get rid of the excess.
  68             V = self.vector().parent()
  69             return V.span( (self**d).vector() for d in xrange(V.dimension()) )
  70
  71
  72         def degree(self):
  73             """
  74             Compute the degree of this element the straightforward way
  75             according to the definition; by appending powers to a list
  76             and figuring out its dimension (that is, whether or not
  77             they're linearly dependent).
  78
  79             EXAMPLES::
  80
  81                 sage: J = eja_ln(4)
  82                 sage: J.one().degree()
  83                 1
  84                 sage: e0,e1,e2,e3 = J.gens()
  85                 sage: (e0 - e1).degree()
  86                 2
  87
  88             """
  89             return self.span_of_powers().dimension()
  90
  91
  92         def minimal_polynomial(self):
  93             return self.matrix().minimal_polynomial()
  94
  95         def characteristic_polynomial(self):
  96             return self.matrix().characteristic_polynomial()
  97
  98
  99 def eja_rn(dimension, field=QQ):
 100     """
 101     Return the Euclidean Jordan Algebra corresponding to the set
 102     `R^n` under the Hadamard product.
 103
 104     EXAMPLES:
 105
 106     This multiplication table can be verified by hand::
 107
 108         sage: J = eja_rn(3)
 109         sage: e0,e1,e2 = J.gens()
 110         sage: e0*e0
 111         e0
 112         sage: e0*e1
 113         0
 114         sage: e0*e2
 115         0
 116         sage: e1*e1
 117         e1
 118         sage: e1*e2
 119         0
 120         sage: e2*e2
 121         e2
 122
 123     """
 124     # The FiniteDimensionalAlgebra constructor takes a list of
 125     # matrices, the ith representing right multiplication by the ith
 126     # basis element in the vector space. So if e_1 = (1,0,0), then
 127     # right (Hadamard) multiplication of x by e_1 picks out the first
 128     # component of x; and likewise for the ith basis element e_i.
 129     Qs = [ matrix(field, dimension, dimension, lambda k,j: 1*(k == j == i))
 130            for i in xrange(dimension) ]
 131
 132     return FiniteDimensionalEuclideanJordanAlgebra(field,Qs)
 133
 134
 135 def eja_ln(dimension, field=QQ):
 136     """
 137     Return the Jordan algebra corresponding to the Lorentz "ice cream"
 138     cone of the given ``dimension``.
 139
 140     EXAMPLES:
 141
 142     This multiplication table can be verified by hand::
 143
 144         sage: J = eja_ln(4)
 145         sage: e0,e1,e2,e3 = J.gens()
 146         sage: e0*e0
 147         e0
 148         sage: e0*e1
 149         e1
 150         sage: e0*e2
 151         e2
 152         sage: e0*e3
 153         e3
 154         sage: e1*e2
 155         0
 156         sage: e1*e3
 157         0
 158         sage: e2*e3
 159         0
 160
 161     In one dimension, this is the reals under multiplication::
 162
 163       sage: J1 = eja_ln(1)
 164       sage: J2 = eja_rn(1)
 165       sage: J1 == J2
 166       True
 167
 168     """
 169     Qs = []
 170     id_matrix = identity_matrix(field,dimension)
 171     for i in xrange(dimension):
 172         ei = id_matrix.column(i)
 173         Qi = zero_matrix(field,dimension)
 174         Qi.set_row(0, ei)
 175         Qi.set_column(0, ei)
 176         Qi += diagonal_matrix(dimension, [ei[0]]*dimension)
 177         # The addition of the diagonal matrix adds an extra ei[0] in the
 178         # upper-left corner of the matrix.
 179         Qi[0,0] = Qi[0,0] * ~field(2)
 180         Qs.append(Qi)
 181
 182     return FiniteDimensionalEuclideanJordanAlgebra(field,Qs)