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.all import *
   9
  10 def eja_rn(dimension, field=QQ):
  11     """
  12     Return the Euclidean Jordan Algebra corresponding to the set
  13     `R^n` under the Hadamard product.
  14
  15     EXAMPLES:
  16
  17     This multiplication table can be verified by hand::
  18
  19         sage: J = eja_rn(3)
  20         sage: e0,e1,e2 = J.gens()
  21         sage: e0*e0
  22         e0
  23         sage: e0*e1
  24         0
  25         sage: e0*e2
  26         0
  27         sage: e1*e1
  28         e1
  29         sage: e1*e2
  30         0
  31         sage: e2*e2
  32         e2
  33
  34     """
  35     # The FiniteDimensionalAlgebra constructor takes a list of
  36     # matrices, the ith representing right multiplication by the ith
  37     # basis element in the vector space. So if e_1 = (1,0,0), then
  38     # right (Hadamard) multiplication of x by e_1 picks out the first
  39     # component of x; and likewise for the ith basis element e_i.
  40     Qs = [ matrix(field, dimension, dimension, lambda k,j: 1*(k == j == i))
  41            for i in xrange(dimension) ]
  42     A = FiniteDimensionalAlgebra(QQ,Qs,assume_associative=True)
  43     return JordanAlgebra(A)
  44
  45
  46 def eja_ln(dimension, field=QQ):
  47     """
  48     Return the Jordan algebra corresponding to the Lorenzt "ice cream"
  49     cone of the given ``dimension``.
  50
  51     EXAMPLES:
  52
  53     This multiplication table can be verified by hand::
  54
  55         sage: J = eja_ln(4)
  56         sage: e0,e1,e2,e3 = J.gens()
  57         sage: e0*e0
  58         e0
  59         sage: e0*e1
  60         e1
  61         sage: e0*e2
  62         e2
  63         sage: e0*e3
  64         e3
  65         sage: e1*e2
  66         0
  67         sage: e1*e3
  68         0
  69         sage: e2*e3
  70         0
  71
  72     In one dimension, this is the reals under multiplication::
  73
  74       sage: J1 = eja_ln(1)
  75       sage: J2 = eja_rn(1)
  76       sage: J1 == J2
  77       True
  78
  79     """
  80     Qs = []
  81     id_matrix = identity_matrix(field,dimension)
  82     for i in xrange(dimension):
  83         ei = id_matrix.column(i)
  84         Qi = zero_matrix(field,dimension)
  85         Qi.set_row(0, ei)
  86         Qi.set_column(0, ei)
  87         Qi += diagonal_matrix(dimension, [ei[0]]*dimension)
  88         # The addition of the diagonal matrix adds an extra ei[0] in the
  89         # upper-left corner of the matrix.
  90         Qi[0,0] = Qi[0,0] * ~field(2)
  91         Qs.append(Qi)
  92
  93     A = FiniteDimensionalAlgebra(QQ,Qs,assume_associative=True)
  94     return JordanAlgebra(A)