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)